A complete resource for finance students, this textbook presents the most common empirical approaches in finance in a co

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Introductory Econometrics for Finance

This bestselling and thoroughly classroom-tested textbook is a complete resource for finance students. A comprehensive and illustrated discussion of the most common empirical approaches in finance prepares students for using econometrics in practice, while detailed case studies help them understand how the techniques are used in relevant financial contexts. Learning outcomes, key concepts and end-of-chapter review questions (with full solutions online) highlight the main chapter takeaways and allow students to self-assess their understanding. Building on the successful dataand problem-driven approach of previous editions, this fourth edition has been updated with new examples, additional introductory material on mathematics and dealing with data, as well as more advanced material on extreme value theory, the generalised method of moments and state space models. A dedicated website, with numerous student and instructor resources including videos and a set of companion manuals for various statistical software – all available free of charge – completes the learning package. is Professor of Finance at the ICMA Centre, Henley Business School, University of Reading, UK where he also obtained his PhD. Chris has diverse research interests and has published over a hundred articles in leading academic and practitioner journals, and six books. He is Associate Editor of several journals, including the Journal of Business Finance and Accounting and the British Accounting Review. He acts as consultant and advisor for various banks, corporations and regulatory and professional bodies in the fields of finance, real estate and econometrics. CHRIS BROOKS

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Introductory Econometrics for Finance FOURTH EDITION CHRIS BROOKS The ICMA Centre, Henley Business School, University of Reading

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University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108422536 DOI: 10.1017/9781108524872 © Chris Brooks 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2002 Second edition published 2008 Third edition published 2014 Fourth edition published 2019 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Brooks, Chris, 1971– author. Title: Introductory econometrics for finance / Chris Brooks, The ICMA Centre, Henley Business School, University of Reading. Description: Fourth edition. | Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2019. | Includes bibliographical references and index. Identifiers: LCCN 2018061692 | ISBN 9781108422536 (hardback : alk. paper) |

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ISBN 9781108436823 (pbk. : alk. paper) Subjects: LCSH: Finance–Econometric models. | Econometrics. Classification: LCC HG173 .B76 2019 | DDC 332.01/5195–dc23 LC record available at https://lccn.loc.gov/2018061692 ISBN 978-1-108-42253-6 Hardback ISBN 978-1-108-43682-3 Paperback Additional resources for this publication at www.cambridge.org/brooks4 Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

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Contents in Brief

List of Figures List of Tables List of Boxes List of Screenshots Preface to the Fourth Edition Acknowledgements Outline of the Remainder of this Book Chapter 1

Introduction and Mathematical Foundations

Chapter 2

Statistical Foundations and Dealing with Data

Chapter 3

A Brief Overview of the Classical Linear Regression Model

Chapter 4

Further Development and Analysis of the Classical Linear Regression Model

Chapter 5

Classical Linear Regression Model Assumptions and Diagnostic Tests

Chapter 6

Univariate Time-Series Modelling and Forecasting

Chapter 7

Multivariate Models

Chapter 8

Modelling Long-Run Relationships in Finance

Chapter 9

Modelling Volatility and Correlation

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Chapter 10

Switching and State Space Models

Chapter 11

Panel Data

Chapter 12

Limited Dependent Variable Models

Chapter 13

Simulation Methods

Chapter 14

Additional Econometric Techniques for Financial Research

Chapter 15

Conducting Empirical Research or Doing a Project or Dissertation in Finance

Appendix Sources of Data Used in This Book and the Accompanying 1 Software Manuals Appendix Tables of Statistical Distributions 2 Glossary References Index

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Detailed Contents

List of Figures List of Tables List of Boxes List of Screenshots Preface to the Fourth Edition Acknowledgements Outline of the Remainder of this Book

Chapter 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

What is Econometrics? Is Financial Econometrics Different? Steps Involved in Formulating an Econometric Model Points to Consider When Reading Articles Functions Differential Calculus Matrices

Chapter 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Statistical Foundations and Dealing with Data

Probability and Probability Distributions A Note on Bayesian versus Classical Statistics Descriptive Statistics Types of Data and Data Aggregation Arithmetic and Geometric Series Future Values and Present Values Returns in Financial Modelling Portfolio Theory Using Matrix Algebra

Chapter 3 3.1

Introduction and Mathematical Foundations

A Brief Overview of the Classical Linear Regression Model

What is a Regression Model? 8

3.2 Regression versus Correlation 3.3 Simple Regression 3.4 Some Further Terminology 3.5 The Assumptions Underlying the Model 3.6 Properties of the OLS Estimator 3.7 Precision and Standard Errors 3.8 An Introduction to Statistical Inference 3.9 A Special Type of Hypothesis Test 3.10 An Example of a Simple t-test of a Theory 3.11 Can UK Unit Trust Managers Beat the Market? 3.12 The Overreaction Hypothesis 3.13 The Exact Significance Level Appendix 3.1 Mathematical Derivations of CLRM Results

Chapter 4

Further Development and Analysis of the Classical Linear Regression Model

4.1 Generalising the Simple Model 4.2 The Constant Term 4.3 How are the Parameters Calculated? 4.4 Testing Multiple Hypotheses: The F-test 4.5 Data Mining and the True Size of the Test 4.6 Qualitative Variables 4.7 Goodness of Fit Statistics 4.8 Hedonic Pricing Models 4.9 Tests of Non-Nested Hypotheses 4.10 Quantile Regression Appendix 4.1 Mathematical Derivations of CLRM Results Appendix 4.2 A Brief Introduction to Factor Models and Principal Components Analysis

Chapter 5

Classical Linear Regression Model Assumptions and Diagnostic Tests

5.1 5.2 5.3

Introduction Statistical Distributions for Diagnostic Tests Assumption (1): E(ut) = 0

5.4

Assumption (2): var(ut) = σ2 < ∞ 9

5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15

Assumption (3): cov(ui, uj) = 0 for i ≠ j Assumption (4): The xt are Non-Stochastic Assumption (5): The Disturbances are Normally Distributed Multicollinearity Adopting the Wrong Functional Form Omission of an Important Variable Inclusion of an Irrelevant Variable Parameter Stability Tests Measurement Errors A Strategy for Constructing Econometric Models Determinants of Sovereign Credit Ratings

Chapter 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

Introduction Some Notation and Concepts Moving Average Processes Autoregressive Processes The Partial Autocorrelation Function ARMA Processes Building ARMA Models: The Box–Jenkins Approach Examples of Time-Series Modelling in Finance Exponential Smoothing Forecasting in Econometrics

Chapter 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

Univariate Time-Series Modelling and Forecasting

Multivariate Models

Motivations Simultaneous Equations Bias So how can Simultaneous Equations Models be Validly Estimated? Can the Original Coefficients be Retrieved from the π s? Simultaneous Equations in Finance A Definition of Exogeneity Triangular Systems Estimation Procedures for Simultaneous Equations Systems An Application of a Simultaneous Equations Approach 10

7.10 7.11 7.12 7.13 7.14 7.15

Vector Autoregressive Models Does the VAR Include Contemporaneous Terms? Block Significance and Causality Tests VARs with Exogenous Variables Impulse Responses and Variance Decompositions VAR Model Example: The Interaction Between Property Returns and the Macroeconomy 7.16 A Couple of Final Points on VARs

Chapter 8

Modelling Long-Run Relationships in Finance

8.1 8.2 8.3 8.4 8.5

Stationarity and Unit Root Testing Tests for Unit Roots in the Presence of Structural Breaks Cointegration Equilibrium Correction or Error Correction Models Testing for Cointegration in Regression: A Residuals-Based Approach 8.6 Methods of Parameter Estimation in Cointegrated Systems 8.7 Lead–Lag and Long-Term Relationships Between Spot and Futures Markets 8.8 Testing for and Estimating Cointegration in Systems 8.9 Purchasing Power Parity 8.10 Cointegration Between International Bond Markets 8.11 Testing the Expectations Hypothesis of the Term Structure of Interest Rates

Chapter 9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10

Modelling Volatility and Correlation

Motivations: An Excursion into Non-Linearity Land Models for Volatility Historical Volatility Implied Volatility Models Exponentially Weighted Moving Average Models Autoregressive Volatility Models Autoregressive Conditionally Heteroscedastic (ARCH) Models Generalised ARCH (GARCH) Models Estimation of ARCH/GARCH Models Extensions to the Basic GARCH Model 11

9.11 Asymmetric GARCH Models 9.12 The GJR model 9.13 The EGARCH Model 9.14 Tests for Asymmetries in Volatility 9.15 GARCH-in-Mean 9.16 Uses of GARCH-Type Models 9.17 Testing Non-Linear Restrictions 9.18 Volatility Forecasting: Some Examples and Results 9.19 Stochastic Volatility Models Revisited 9.20 Forecasting Covariances and Correlations 9.21 Covariance Modelling and Forecasting in Finance 9.22 Simple Covariance Models 9.23 Multivariate GARCH Models 9.24 Direct Correlation Models 9.25 Extensions to the Basic Multivariate GARCH Model 9.26 A Multivariate GARCH Model for the CAPM 9.27 Estimating a Time-Varying Hedge Ratio 9.28 Multivariate Stochastic Volatility Models Appendix 9.1 Parameter Estimation Using Maximum Likelihood

Chapter 10 Switching and State Space Models 10.1 Motivations 10.2 Seasonalities in Financial Markets 10.3 Modelling Seasonality in Financial Data 10.4 Estimating Simple Piecewise Linear Functions 10.5 Markov Switching Models 10.6 A Markov Switching Model for the Real Exchange Rate 10.7 A Markov Switching Model for the Gilt–Equity Yield Ratio 10.8 Threshold Autoregressive Models 10.9 Estimation of Threshold Autoregressive Models 10.10 Specification Tests 10.11 A SETAR Model for the French franc–German mark Exchange Rate 10.12 Threshold Models for FTSE Spot and Futures 10.13 Regime Switching Models and Forecasting 10.14 State Space Models and the Kalman Filter 12

Chapter 11 Panel Data 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9

Introduction: What Are Panel Techniques? What Panel Techniques Are Available? The Fixed Effects Model Time-Fixed Effects Models Investigating Banking Competition The Random Effects Model Panel Data Application to Credit Stability of Banks Panel Unit Root and Cointegration Tests Further Feading

Chapter 12 Limited Dependent Variable Models 12.1 Introduction and Motivation 12.2 The Linear Probability Model 12.3 The Logit Model 12.4 Using a Logit to Test the Pecking Order Hypothesis 12.5 The Probit Model 12.6 Choosing Between the Logit and Probit Models 12.7 Estimation of Limited Dependent Variable Models 12.8 Goodness of Fit Measures for Linear Dependent Variable Models 12.9 Multinomial Linear Dependent Variables 12.10 The Pecking Order Hypothesis Revisited 12.11 Ordered Response Linear Dependent Variables Models 12.12 Are Unsolicited Credit Ratings Biased Downwards? An Ordered Probit Analysis 12.13 Censored and Truncated Dependent Variables Appendix 12.1 The Maximum Likelihood Estimator for Logit and Probit Models

Chapter 13 Simulation Methods 13.1 13.2 13.3 13.4 13.5 13.6

Motivations Monte Carlo Simulations Variance Reduction Techniques Bootstrapping Random Number Generation Disadvantages of the Simulation Approach 13

13.7 An Example of Monte Carlo Simulation 13.8 An Example of how to Simulate the Price of a Financial Option 13.9 An Example of Bootstrapping to Calculate Capital Risk Requirements

Chapter 14 Additional Econometric Techniques for Financial Research 14.1 14.2 14.3 14.4

Event Studies Tests of the CAPM and the Fama–French Methodology Extreme Value Theory The Generalised Method of Moments

Chapter 15 Conducting Empirical Research or Doing a Project or Dissertation in Finance 15.1 What is an Empirical Research Project? 15.2 Selecting the Topic 15.3 Sponsored or Independent Research? 15.4 The Research Proposal 15.5 Working Papers and Literature on the Internet 15.6 Getting the Data 15.7 Choice of Computer Software 15.8 Methodology 15.9 How Might the Finished Project Look? 15.10 Presentational Issues

Appendix 1 Sources of Data Used in This Book and the Accompanying Software Manuals Appendix 2 Tables of Statistical Distributions Glossary References Index

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Figures

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

Steps involved in formulating an econometric model A plot of hours studied (x) against grade-point average (y) Examples of different straight line graphs Example of a general polynomial function Examples of quadratic functions A plot of an exponential function A plot of a logarithmic function The tangents to a curve y = f(x), its first derivative and its second derivative around the point x = −6 The probability distribution function for the sum of two dice The pdf for a normal distribution The cdf for a normal distribution A normal versus a skewed distribution A normal versus a leptokurtic distribution A time-series plot and scatter plot of the performance of two fund managers Scatter plot of two variables, y and x Scatter plot of two variables with a line of best fit chosen by eye Method of OLS fitting a line to the data by minimising the sum of squared residuals Plot of a single observation, together with the line of best fit, the residual and the fitted value How RSS varies with different values of β Scatter plot of excess returns on fund XXX versus excess returns on the market portfolio No observations close to the y-axis The bias versus variance trade-off when selecting between estimators 15

3.9 3.10 3.11 3.12 3.13 3.14 3.15

Effect on the standard errors of the coefficient estimates when are narrowly dispersed Effect on the standard errors of the coefficient estimates when are widely dispersed Effect on the standard errors of large Effect on the standard errors of small The t-distribution versus the normal Rejection regions for a two-sided 5% hypothesis test Rejection region for a one-sided hypothesis test of the form H0:β = β*, H1:β < β*

3.16 Rejection region for a one-sided hypothesis test of the form H0:β = β*, H1:β > β* 3.17 Critical values and rejection regions for a t20;5% 3.18 Frequency distribution of t-ratios of mutual fund alphas (gross of transactions costs) 3.19 Frequency distribution of t-ratios of mutual fund alphas (net of transactions costs) 3.20 Performance of UK unit trusts, 1979–2000 4.1 R2 = 0 demonstrated by a flat estimated line, i.e., a zero slope coefficient 4.2 R2 = 1 when all data points lie exactly on the estimated line 5.1 Effect of no intercept on a regression line 5.2 Graphical illustration of heteroscedasticity 5.3 Plot of against showing positive autocorrelation 5.4 Plot of over time, showing positive autocorrelation 5.5 Plot of against showing negative autocorrelation 5.6 Plot of over time, showing negative autocorrelation 5.7 Plot of against showing no autocorrelation 5.8 Plot of over time, showing no autocorrelation 5.9 Rejection and non-rejection regions for DW test 5.10 Regression residuals from stock return data, showing large outlier for October 1987 5.11 Possible effect of an outlier on OLS estimation 5.12 Relationship between y and x2 in a quadratic regression for different values of β2 and β3 16

5.13 Plot of a variable showing suggestion for break date 6.1 Autocorrelation function for sample MA(2) process 6.2 Sample autocorrelation and partial autocorrelation functions for an MA(1) model: yt = −0.5ut−1 + ut 6.3 Sample autocorrelation and partial autocorrelation functions for an MA(2) model: yt = 0.5ut−1 − 0.25ut−2 + ut 6.4 Sample autocorrelation and partial autocorrelation functions for a slowly decaying AR(1) model: yt = 0.9yt−1 + ut 6.5 Sample autocorrelation and partial autocorrelation functions for a more rapidly decaying AR(1) model: yt = 0.5yt−1 + ut 6.6 Sample autocorrelation and partial autocorrelation functions for a more rapidly decaying AR(1) model with negative coefficient: yt = −0.5yt−1 + ut 6.7 6.8 6.9 7.1 7.2 8.1 8.2

8.3 8.4 8.5 8.6 9.1 9.2 9.3

Sample autocorrelation and partial autocorrelation functions for a non-stationary model (i.e., a unit coefficient): yt = yt−1 + ut Sample autocorrelation and partial autocorrelation functions for an ARMA(1, 1) model: yt = 0.5yt−1 + 0.5ut−1 + ut Use of in-sample and out-of-sample periods for analysis Impulse responses and standard error bands for innovations in unexpected inflation equation errors Impulse responses and standard error bands for innovations in the dividend yields Value of R2 for 1000 sets of regressions of a non-stationary variable on another independent non-stationary variable Value of t-ratio of slope coefficient for 1,000 sets of regressions of a non-stationary variable on another independent nonstationary variable Example of a white noise process Time-series plot of a random walk versus a random walk with drift Time-series plot of a deterministic trend process Autoregressive processes with differing values of ϕ (0, 0.8, 1) Daily S&P returns for August 2003–July 2018 The problem of local optima in maximum likelihood estimation News impact curves for S&P500 returns using coefficients implied from GARCH and GJR model estimates 17

9.4 9.5 10.1 10.2 10.3 10.4 10.5 10.6 12.1 12.2 12.3 14.1

Three approaches to hypothesis testing under maximum likelihood Time-varying hedge ratios derived from symmetric and asymmetric BEKK models for FTSE returns Sample time-series plot illustrating a regime shift Use of intercept dummy variables for quarterly data Use of slope dummy variables Piecewise linear model with threshold x* Unconditional distribution of US GEYR together with a normal distribution with the same mean and variance Value of GEYR and probability that it is in the high GEYR regime for the UK The fatal flaw of the linear probability model The logit model Modelling charitable donations as a function of income Pdfs for the Weibull, Gumbel and Frechét distributions

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Tables

Sample data on hours of study and grades Annual performance of two funds Impact of different compounding frequencies on the effective interest rate and terminal value of an investment 2.3 How to construct a series in real terms from a nominal one 3.1 Sample data on fund XXX to motivate OLS estimation 3.2 Critical values from the standard normal versus t-distribution 3.3 Classifying hypothesis testing errors and correct conclusions 3.4 Summary statistics for the estimated regression results for equation (3.34) 3.5 Summary statistics for unit trust returns, January 1979–May 2000 3.6 CAPM regression results for unit trust returns, January 1979– May 2000 3.7 Is there an overreaction effect in the UK stock market? 4.1 Hedonic model of rental values in Quebec City, 1990. Dependent variable: Canadian dollars per month 4.2 OLS and quantile regression results for the Magellan fund 4A.1 Principal component ordered eigenvalues for Dutch interest rates, 1962–70 4A.2 Factor loadings of the first and second principal components for Dutch interest rates, 1962–70 5.1 Constructing a series of lagged values and first differences 5.2 Determinants and impacts of sovereign credit ratings 5.3 Do ratings add to public information? 5.4 What determines reactions to ratings announcements? 6.1 Uncovered interest parity test results 6.2 Forecast error aggregation 7.1 Call bid–ask spread and trading volume regression 7.2 Put bid–ask spread and trading volume regression 1.1 2.1 2.2

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7.3 7.4 7.5 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 9.1 9.2 9.3 9.4 9.5 10.1 10.2 10.3 10.4 10.5 10.6 10.7

Granger causality tests and implied restrictions on VAR models Marginal significance levels associated with joint F-tests Variance decompositions for the property sector index residuals Critical values for DF tests (Fuller, 1976, p. 373) Recursive unit root tests for interest rates allowing for structural breaks DF tests on log-prices and returns for high frequency FTSE data Estimated potentially cointegrating equation and test for cointegration for high frequency FTSE data Estimated error correction model for high frequency FTSE data Comparison of out-of-sample forecasting accuracy Trading profitability of the error correction model with cost of carry Cointegration tests of PPP with European data DF tests for international bond indices Cointegration tests for pairs of international bond indices Johansen tests for cointegration between international bond yields Variance decompositions for VAR of international bond yields Impulse responses for VAR of international bond yields Tests of the expectations hypothesis using the US zero coupon yield curve with monthly data GARCH versus implied volatility EGARCH versus implied volatility Out-of-sample predictive power for weekly volatility forecasts Comparisons of the relative information content of out-of-sample volatility forecasts Hedging effectiveness: summary statistics for portfolio returns Values and significances of days of the week coefficients Day-of-the-week effects with the inclusion of interactive dummy variables with the risk proxy Estimates of the Markov switching model for real exchange rates Estimated parameters for the Markov switching models SETAR model for FRF–DEM FRF–DEM forecast accuracies Linear AR(3) model for the basis 20

10.8 A two-threshold SETAR model for the basis 10.9 Unit trust performance with time-varying beta estimation 11.1 Tests of banking market equilibrium with fixed effects panel models 11.2 Tests of competition in banking with fixed effects panel models 11.3 Results of random effects panel regression for credit stability of Central and East European banks 11.4 Panel unit root test results for economic growth and financial development 11.5 Panel cointegration test results for economic growth and financial development 12.1 Logit estimation of the probability of external financing 12.2 Multinomial logit estimation of the type of external financing 12.3 Ordered probit model results for the determinants of credit ratings 12.4 Two-step ordered probit model allowing for selectivity bias in the determinants of credit ratings 13.1 EGARCH estimates for currency futures returns 13.2 Autoregressive volatility estimates for currency futures returns 13.3 Minimum capital risk requirements for currency futures as a percentage of the initial value of the position 14.1 Fama and MacBeth’s results on testing the CAPM 14.2 Threshold percentage returns, corresponding empirical quantiles and the number of exceedences 14.3 Maximum likelihood estimates of the parameters of the generalised Pareto distribution 14.4 Models that predict the actual left tail quantile most accurately 14.5 GMM estimates of the effect of stock markets and bank lending on economic growth 15.1 Journals in finance and econometrics 15.2 Useful internet sites for financial literature 15.3 Suggested structure for a typical dissertation or project A2.1 Normal critical values for different values of α A2.2 Critical values of Student’s t-distribution for different probability levels, α and degrees of freedom, ν A2.3 Upper 5% critical values for F-distribution A2.4 Upper 1% critical values for F-distribution 21

A2.5 Chi-squared critical values for different values of α and degrees of freedom, υ A2.6 Lower and upper 1% critical values for the Durbin–Watson statistic A2.7 Dickey–Fuller critical values for different significance levels, α A2.8 Critical values for the Engle–Granger cointegration test on regression residuals with no constant in test regression A2.9 Quantiles of the asymptotic distribution of the Johansen cointegration rank test statistics (constant in cointegrating vectors only) A2.10 Quantiles of the asymptotic distribution of the Johansen cointegration rank test statistics (constant, i.e., a drift only in VAR and in cointegrating vector) A2.11 Quantiles of the asymptotic distribution of the Johansen cointegration rank test statistics (constant in cointegrating vector and VAR, trend in cointegrating vector)

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Boxes

Examples of the uses of econometrics Points to consider when reading a published paper The roots of a quadratic equation Manipulating powers and their indices The laws of logs The population and the sample Time-series data Log returns Names for y and xs in regression models Reasons for the inclusion of the disturbance term Assumptions concerning disturbance terms and their interpretation 3.4 Standard error estimators 3.5 Conducting a test of significance 3.6 Carrying out a hypothesis test using confidence intervals 3.7 The test of significance and confidence interval approaches compared 3.8 Type I and type II errors 3.9 Reasons for stock market overreactions 3.10 Ranking stocks and forming portfolios 3.11 Portfolio monitoring 4.1 The relationship between the regression F-statistic and R2 4.2 Selecting between models 5.1 Conducting White’s test 5.2 ‘Solutions’ for Heteroscedasticity 5.3 Conditions for DW to be a valid test 5.4 Conducting a Breusch–Godfrey test 5.5 The Cochrane–Orcutt procedure 1.1 1.2 1.3 1.4 1.5 2.1 2.2 2.3 3.1 3.2 3.3

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5.6 5.7 6.1 6.2 6.3 7.1 7.2 7.3 8.1 8.2 9.1 9.2 9.3 10.1 10.2 11.1 12.1 12.2 13.1 13.2 13.3 13.4 13.5 13.6 14.1 15.1

Observations for the dummy variable Conducting a Chow test The stationarity condition for an AR(p) model The invertibility condition for an MA(2) model Naive forecasting methods Determining whether an equation is identified Conducting a Hausman test for exogeneity Forecasting with VARs Stationarity tests Multiple cointegrating relationships Testing for ‘ARCH effects’ Estimating an ARCH or GARCH model Using maximum likelihood estimation in practice How do dummy variables work? Parameter estimation using the Kalman filter Fixed or random effects? Parameter interpretation for probit and logit models The differences between censored and truncated dependent variables Conducting a Monte Carlo simulation Re-sampling the data Re-sampling from the residuals Setting up a Monte Carlo simulation Simulating the price of an option Generating draws from a GARCH process The three generalised extreme value distributions Possible types of research project

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Screenshots

2.1 2.2 2.3 2.4 2.5

Setting up a variance–covariance matrix in Excel The spreadsheet for constructing the efficient frontier Completing the Solver window A plot of the completed efficient frontier The capital market line and efficient frontier

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Preface to the Fourth Edition

All of the motivations for the first edition, described below, seem just as important today. Given that the book seems to have gone down well with readers, I have left the style largely unaltered but added a lot of new material. The main motivations for writing the first edition of the book were: To write a book that focused on using and applying the techniques rather than deriving proofs and learning formulae. To write an accessible textbook that required no prior knowledge of econometrics, but which also covered more recently developed approaches usually only found in more advanced texts. To use examples and terminology from finance rather than economics since there are many introductory texts in econometrics aimed at students of economics but none for students of finance. To populate the book with case studies of the use of econometrics in practice taken from the academic finance literature. To include sample instructions, screen dumps and computer output from a popular econometrics package. This enabled readers to see how the techniques can be implemented in practice. In this fourth edition, the EViews instructions have been separated off and are available free of charge on the book’s web site along with parallel manuals for other packages including Stata, Python and R. To develop a companion web site containing answers to end of chapter questions, a multiple choice question bank with feedback, PowerPoint slides and other supporting materials. What is New in the Fourth Edition The fourth edition includes a number of important new features (1) Students of finance have enormously varying backgrounds, and in particular varying levels of training in elementary mathematics and 26

statistics. In order to make the book more self-contained, the introductory chapter has again been expanded. So the material previously in Chapter 2 has been separated into introductory maths (Chapter 1) and introductory statistics/dealing with data (Chapter 2). (2) More new material has been added on state space models and their estimation using the Kalman filter in Chapter 10. (3) A chapter has been added which collects together a number of techniques often used in financial research, including event studies and the Fama MacBeth approach (previously elsewhere in the book) and new sections on using extreme value distribution to model the fat tails in financial series and on estimating models with the generalised method of moments. (4) The incorporation of EViews directly into the core of the book may have been a distraction for those using other packages. Thus, as stated above, in the new edition the EViews instructions have been separated off and are available free of charge on the book’s web site along with parallel manuals for other packages including Stata, Python and R. This package should ensure that the book fits the bill whatever the reader’s preferred software. Motivations for the First Edition This book had its genesis in two sets of lectures given annually by the author at the ICMA Centre (formerly the ISMA Centre), Henley Business School, University of Reading and arose partly from several years of frustration at the lack of an appropriate textbook. In the past, finance was but a small sub-discipline drawn from economics and accounting, and therefore it was generally safe to assume that students of finance were well grounded in economic principles; econometrics would be taught using economic motivations and examples. However, finance as a subject has taken on a life of its own in recent years. Drawn in by perceptions of exciting careers in the financial markets, the number of students of finance has grown phenomenally all around the world. At the same time, the diversity of educational backgrounds of students taking finance courses has also expanded. It is not uncommon to find undergraduate students of finance even without advanced high-school qualifications in mathematics or economics. Conversely, many with PhDs in physics or engineering are also attracted to study finance at the Masters level. Unfortunately, authors of textbooks failed to keep pace with the 27

change in the nature of students. In my opinion, the currently available textbooks fall short of the requirements of this market in three main regards, which this book seeks to address (1) Books fall into two distinct and non-overlapping categories: the introductory and the advanced. Introductory textbooks are at the appropriate level for students with limited backgrounds in mathematics or statistics, but their focus is too narrow. They often spend too long deriving the most basic results, and treatment of important, interesting and relevant topics (such as simulations methods, VAR modelling, etc.) is covered in only the last few pages, if at all. The more advanced textbooks, meanwhile, usually require a quantum leap in the level of mathematical ability assumed of readers, so that such books cannot be used on courses lasting only one or two semesters, or where students have differing backgrounds. In this book, I have tried to sweep a broad brush over a large number of different econometric techniques that are relevant to the analysis of financial and other data. (2) Many of the currently available textbooks with broad coverage are too theoretical in nature and students can often, after reading such a book, still have no idea of how to tackle real-world problems themselves, even if they have mastered the techniques in theory. This book and the accompanying software manuals should assist students who wish to learn how to estimate models for themselves – for example, if they are required to complete a project or dissertation. Some examples have been developed especially for this book, while many others are drawn from the academic finance literature. In my opinion, this is an essential but rare feature of a textbook that should help to show students how econometrics is really applied. It is also hoped that this approach will encourage some students to delve deeper into the literature, and will give useful pointers and stimulate ideas for research projects. It should, however, be stated at the outset that the purpose of including examples from the academic finance print is not to provide a comprehensive overview of the literature or to discuss all of the relevant work in those areas, but rather to illustrate the techniques. Therefore, the literature reviews may be considered deliberately deficient, with interested readers directed to the suggested readings and the references therein. (3) With few exceptions, almost all textbooks that are aimed at the introductory level draw their motivations and examples from 28

economics, which may be of limited interest to students of finance or business. To see this, try motivating regression relationships using an example such as the effect of changes in income on consumption and watch your audience, who are primarily interested in business and finance applications, slip away and lose interest in the first ten minutes of your course. Who Should Read this Book? The intended audience is undergraduates or Masters/MBA and PhD students who require a broad knowledge of modern econometric techniques commonly employed in the finance literature. It is hoped that the book will also be useful for researchers (both academics and practitioners), who require an introduction to the statistical tools commonly employed in the area of finance. The book can be used for courses covering financial time-series analysis or financial econometrics in undergraduate or post-graduate programmes in finance, financial economics, securities and investments. Although the applications and motivations for model-building given in the book are drawn from finance, the empirical testing of theories in many other disciplines, such as management studies, business studies, real estate, economics and so on, may usefully employ econometric analysis. For this group, the book may also prove useful. Finally, while the present text is designed mainly for students at the undergraduate or Masters level, it could also provide introductory reading in financial modelling for finance doctoral programmes where students have backgrounds which do not include courses in modern econometric techniques. Pre-Requisites for Good Understanding of This Material In order to make the book as accessible as possible, no prior knowledge of statistics, econometrics or algebra is required, although those with a prior exposure to calculus, algebra (including matrices) and basic statistics will be able to progress more quickly. The emphasis throughout the book is on a valid application of the techniques to real data and problems in finance. In the finance and investment area, it is assumed that the reader has knowledge of the fundamentals of corporate finance, financial markets and investment. Therefore, subjects such as portfolio theory, the capital asset pricing model (CAPM) and arbitrage pricing theory (APT), the efficient 29

markets hypothesis, the pricing of derivative securities and the term structure of interest rates, which are frequently referred to throughout the book, are not explained from first principles in this text. There are very many good books available in corporate finance, in investments and in futures and options, including those by Brealey and Myers (2013), Bodie, Kane and Marcus (2014) and Hull (2017) respectively.

30

Acknowledgements

I am grateful to Gita Persand, Olan Henry, James Chong and Apostolos Katsaris, who assisted with various parts of the software applications for the first edition. I am also grateful to Hilary Feltham for assistance with Chapters 1 and 2. I would also like to thank Simon Burke, James Chong and Con Keating for detailed and constructive comments on various drafts of the first edition, Simon Burke for suggestions on parts of the second edition, Mike Clements, Jo Cox, Eunyoung Mallet, Ogonna Nneji, Ioannis Oikonomou and Chardan Wese Simen for comments on part of the third edition and Marcel Prokopczuk for comments on part of the fourth edition. I have additionally benefited from the comments, suggestions and questions of the following list of people, many of whom sent useful emails pointing out typos or inaccuracies: Zary Aftab, Panos BallisPapanastasiou, Mirco Balatti, Peter Burridge, Kyongwook Choi, Rishi Chopra, Araceli Ortega Diaz, Xiaoming Ding, Thomas Eilertsen, Waleid Eldien, Junjong Eo, Merlyn Foo, Andrea Gheno, Christopher Gilbert, Kimon Gomozias, Jan de Gooijer and his colleagues, Cherif Guermat, Abid Hameed, Ibrahim Jamali, Kejia Jia, Arty Khemlani, Margaret Lynch, David McCaffrey, Tehri Jokipii, Emese Lazar, Zhao Liuyan, Dimitri Lvov, Bill McCabe, Junshi Ma, Raffaele Mancuso, David Merchan, Yue Min, Victor Murinde, Kyoung Gook Park, Mikael Petitjean, Marcelo Perlin, Thai Pham, Jean-Sebastien Pourchet, Marcel Prokopczuk, Tao Qingmei, Satya Sahoo, Lisa Schopohl, Guilherme Silva, Jerry Sin, Andre-Tudor Stancu, Silvia Stanescu, Fred Sterbenz, Birgit Strikholm, Yiguo Sun, Li Qui, Panagiotis Varlagas, Jakub Vojtek, Henk von Eije, Jue Wang, Robert Wichmann and Meng-Feng Yen. The publisher and author have used their best endeavours to ensure that the URLs for external web sites referred to in this book are correct and active at the time of going to press. However, the publisher and author have no responsibility for the web sites and can make no guarantee that a site will remain live or that the content is or will remain appropriate.

31

Outline of the Remainder of this Book

Chapter 1 This covers the key mathematical techniques that readers will need some familiarity with to be able to get the most out of the remainder of this book. It starts with a discussion of what econometrics is about and how to set up an econometric model, then moves on to present the mathematical material on functions, and powers, exponents and logarithms of numbers. It then proceeds to explain the basics of differentiation and matrix algebra, which is illustrated via the construction of optimal portfolio weights. Chapter 2 This chapter presents the statistical foundations of econometrics and the beginnings of how to work with financial data. It covers key results in statistics, discusses probability distributions, how to summarise data and different types of data. The chapter then moves on to discuss the calculation of present and future values, compounding and discounting, and how to calculate nominal and real returns in various ways. Chapter 3 This introduces the classical linear regression model (CLRM). The ordinary least squares (OLS) estimator is derived and its interpretation discussed. The conditions for OLS optimality are stated and explained. A hypothesis testing framework is developed and examined in the context of the linear model. Examples employed include Jensen’s classic study of mutual fund performance measurement and tests of the ‘overreaction hypothesis’ in the context of the UK stock market. Chapter 4 This continues and develops the material of Chapter 3 by generalising the bivariate model to multiple regression – i.e., models with many variables. The framework for testing multiple hypotheses is outlined, and measures 32

of how well the model fits the data are described. Case studies include modelling rental values and an application of principal components analysis (PCA) to interest rates. Chapter 5 Chapter 5 examines the important but often neglected topic of diagnostic testing. The consequences of violations of the CLRM assumptions are described, along with plausible remedial steps. Model-building philosophies are discussed, with particular reference to the general-tospecific approach. Applications covered in this chapter include the determination of sovereign credit ratings. Chapter 6 This presents an introduction to time-series models, including their motivation and a description of the characteristics of financial data that they can and cannot capture. The chapter commences with a presentation of the features of some standard models of stochastic (white noise, moving average, autoregressive and mixed ARMA) processes. The chapter continues by showing how the appropriate model can be chosen for a set of actual data, how the model is estimated and how model adequacy checks are performed. The generation of forecasts from such models is discussed, as are the criteria by which these forecasts can be evaluated. Examples include model-building for UK house prices, and tests of the exchange rate covered and uncovered interest parity hypotheses. Chapter 7 This extends the analysis from univariate to multivariate models. Multivariate models are motivated by way of explanation of the possible existence of bi-directional causality in financial relationships, and the simultaneous equations bias that results if this is ignored. Estimation techniques for simultaneous equations models are outlined. Vector autoregressive (VAR) models, which have become extremely popular in the empirical finance literature, are also covered. The interpretation of VARs is explained by way of joint tests of restrictions, causality tests, impulse responses and variance decompositions. Relevant examples discussed in this chapter are the simultaneous relationship between bid– ask spreads and trading volume in the context of options pricing, and the 33

relationship between property returns and macroeconomic variables. Chapter 8 The first section of the chapter discusses unit root processes and presents tests for non-stationarity in time-series. The concept of and tests for cointegration, and the formulation of error correction models, are then discussed in the context of both the single equation framework of Engle– Granger, and the multivariate framework of Johansen. Applications studied in Chapter 8 include spot and futures markets, tests for cointegration between international bond markets and tests of the purchasing power parity (PPP) hypothesis and of the expectations hypothesis of the term structure of interest rates. Chapter 9 This covers the important topic of volatility and correlation modelling and forecasting. This chapter starts by discussing in general terms the issue of non-linearity in financial time series. The class of ARCH (autoregressive conditionally heteroscedastic) models and the motivation for this formulation are then discussed. Other models are also presented, including extensions of the basic model such as GARCH, GARCH-M, EGARCH and GJR formulations. Examples of the huge number of applications are discussed, with particular reference to stock returns. Multivariate GARCH and conditional correlation models are described, and applications to the estimation of conditional betas and time-varying hedge ratios, and to financial risk measurement, are given. Chapter 10 This begins by discussing how to test for and model regime shifts or switches of behaviour in financial series that can arise from changes in government policy, market trading conditions or microstructure, among other causes. This chapter then introduces the Markov switching approach to dealing with regime shifts. Threshold autoregression is also discussed, along with issues relating to the estimation of such models. Examples include the modelling of exchange rates within a managed floating environment, modelling and forecasting the gilt–equity yield ratio and models of movements of the difference between spot and futures prices. Finally, the second part of the chapter moves on to examine how to specify 34

models with time-varying parameters using the state space form and how to estimate them with the Kalman filter. Chapter 11 This chapter focuses on how to deal appropriately with longitudinal data – that is, data having both time-series and cross-sectional dimensions. Fixed effect and random effect models are explained and illustrated by way of examples on banking competition in the UK and on credit stability in Central and Eastern Europe. Entity fixed and time-fixed effects models are elucidated and distinguished. Chapter 12 This chapter describes various models that are appropriate for situations where the dependent variable is not continuous. Readers will learn how to construct, estimate and interpret such models, and to distinguish and select between alternative specifications. Examples used include a test of the pecking order hypothesis in corporate finance and the modelling of unsolicited credit ratings. Chapter 13 This presents an introduction to the use of simulations in econometrics and finance. Motivations are given for the use of repeated sampling, and a distinction is drawn between Monte Carlo simulation and bootstrapping. The reader is shown how to set up a simulation, and examples are given in options pricing and financial risk management to demonstrate the usefulness of these techniques. Chapter 14 This chapter presents a collection of techniques that are particularly useful for conducting research in finance. It begins with detailed illustrations of how to conduct event studies, which are commonly used in corporate finance applications, and how to use the Fama-French factor model approach to asset pricing. The chapter then proceeds to present the families of extreme value models that are used to accurately capture the fat tails of asset return distributions and as the basis for value at risk calculations. Finally, the chapter covers the generalised method of moments (GMM) 35

technique, which has become increasingly popular in recent years for estimating a range of different types of models in finance. Chapter 15 This offers suggestions related to conducting a project or dissertation in empirical finance. It introduces the sources of financial and economic data available on the internet and elsewhere, and recommends relevant online information and literature on research in financial markets and financial time series. The chapter also suggests ideas for what might constitute a good structure for a dissertation on this subject, how to generate ideas for a suitable topic, what format the report could take, and some common pitfalls.

36

1 Introduction and Mathematical Foundations

LEARNING OUTCOMES In this chapter, you will learn how to Describe the key steps involved in building an econometric model Work with powers, exponents and logarithms Plot, interpret and calculate the roots of functions Use sigma (Σ) and pi (Π) notation Apply rules to differentiate various types functions Work with matrices Calculate the trace, inverse and eigenvalues of a matrix Construct and interpret utility functions

Learning econometrics is in many ways like learning a new language. To begin with, nothing makes sense and it is as if it is impossible to see through the fog created by all the unfamiliar terminology. While the way of writing the models – the notation – may make the situation appear more complex, in fact it is supposed to achieve the exact opposite. The ideas themselves are mostly not so complicated, it is just a matter of learning enough of the language that everything fits into place. So if you have never studied the subject before, then persevere through this preliminary chapter and you will hopefully be on your way to being fully fluent in econometrics! This chapter comprises two parts. The first sets the scene for the book by discussing in broad terms the questions of what econometrics is, and the kinds of problems that can be tackled using econometrics. The second part 37

of the chapter covers the mathematical techniques that underpin approaches to modelling and dealing with data in finance. Those with some prior background in algebra and introductory mathematics may skip the second part of this chapter without loss of continuity, but hopefully the material will also constitute a useful refresher for those who have studied mathematics but a long time ago!

1.1 What is Econometrics? The literal meaning of the word ‘econometrics’ is ‘measurement in economics’. The first five letters of the word suggest correctly that the origins of econometrics are rooted in economics. However, the main techniques employed for studying economic problems are of equal importance in financial applications. As the term is used in this book, financial econometrics will be defined as the application of statistical techniques to problems in finance. Financial econometrics can be useful for testing theories in finance, determining asset prices or returns, testing hypotheses concerning the relationships between variables, examining the effect on financial markets of changes in economic conditions, forecasting future values of financial variables and for financial decision-making. A list of possible examples of where econometrics may be useful is given in Box 1.1. BOX 1.1 Examples of the uses of econometrics (1) Testing whether financial markets are weak-form informationally efficient (2) Testing whether the capital asset pricing model (CAPM) or arbitrage pricing theory (APT) represent superior models for the determination of returns on risky assets (3) Measuring and forecasting the volatility of bond returns (4) Explaining the determinants of bond credit ratings used by the ratings agencies (5) Modelling long-term relationships between prices and exchange rates (6) Determining the optimal hedge ratio for a spot position in oil (7) Testing technical trading rules to determine which makes the most money (8) Testing the hypothesis that earnings or dividend announcements 38

have no effect on stock prices (9) Testing whether spot or futures markets react more rapidly to news (10) Forecasting the correlation between the stock indices of two countries. The list in Box 1.1 is of course by no means exhaustive, but it hopefully gives some flavour of the usefulness of econometric tools in terms of their financial applicability.

1.2 Is Financial Econometrics Different from ‘Economic Econometrics’? As previously stated, the tools commonly used in financial applications are fundamentally the same as those used in economic applications, although the emphasis and the sets of problems that are likely to be encountered when analysing the two sets of data are somewhat different. Financial data often differ from macroeconomic data in terms of their frequency, accuracy, seasonality and other properties. In economics, a serious problem is often a lack of data at hand for testing the theory or hypothesis of interest – this is sometimes called a ‘small samples problem’. It might be, for example, that data are required on government budget deficits, or population figures, which are measured only on an annual basis. If the methods used to measure these quantities changed a quarter of a century ago, then only at most twenty-five of these annual observations are usefully available. Two other problems that are often encountered in conducting applied econometric work in the arena of economics are those of measurement error and data revisions. These difficulties are simply that the data may be estimated, or measured with error, and will often be subject to several vintages of subsequent revisions. For example, a researcher may estimate an economic model of the effect on national output of investment in computer technology using a set of published data, only to find that the data for the last two years have been revised substantially in the next, updated publication. These issues are usually of less concern in finance. Financial data come in many shapes and forms, but in general the prices and other entities that are recorded are those at which trades actually took place, or which were 39

quoted on the screens of information providers. There exists, of course, the possibility for typos or for the data measurement method to change (for example, owing to stock index re-balancing or re-basing). But in general the measurement error and revisions problems are far less serious in the financial context. Similarly, some sets of financial data are observed at much higher frequencies than macroeconomic data. Asset prices or yields are often available at daily, hourly or minute-by-minute frequencies. Thus the number of observations available for analysis can potentially be very large – perhaps thousands or even millions, making financial data the envy of macro-econometricians! The implication is that more powerful techniques can often be applied to financial than economic data, and that researchers may also have more confidence in the results. Furthermore, the analysis of financial data also brings with it a number of new problems. While the difficulties associated with handling and processing such a large amount of data are not usually an issue given recent and continuing advances in computer power, financial data often have a number of additional characteristics. For example, financial data are often considered very ‘noisy’, which means that it is more difficult to separate underlying trends or patterns from random and uninteresting features. Financial data are also almost always not normally distributed in spite of the fact that most techniques in econometrics assume that they are. High frequency data often contain additional ‘patterns’ which are the result of the way that the market works, or the way that prices are recorded. These features need to be considered in the model-building process, even if they are not directly of interest to the researcher. One of the most rapidly evolving areas of financial application of statistical tools is in the modelling of market microstructure problems. ‘Market microstructure’ may broadly be defined as the process whereby investors’ preferences and desires are translated into financial market transactions. It is evident that microstructure effects are important and represent a key difference between financial and other types of data. These effects can potentially impact on many other areas of finance. For example, market rigidities or frictions can imply that current asset prices do not fully reflect future expected cashflows (see the discussion in Chapter 10 of this book). Also, investors are likely to require compensation for holding securities that are illiquid, and therefore embody a risk that they will be difficult to sell owing to the relatively high probability of a lack of willing purchasers at the time of desired sale. Measures such as volume or the time between trades are sometimes used 40

as proxies for market liquidity. A comprehensive survey of the literature on market microstructure is given by Madhavan (2000). He identifies several aspects of the market microstructure literature, including price formation and price discovery, issues relating to market structure and design, information and disclosure. There are also relevant books by O’Hara (1995), Harris (2002) and Hasbrouck (2007). At the same time, there has been considerable advancement in the sophistication of econometric models applied to microstructure problems. For example, an important innovation was the auto-regressive conditional duration (ACD) model attributed to Engle and Russell (1998). An interesting application can be found in Dufour and Engle (2000), who examine the effect of the time between trades on the price-impact of the trade and the speed of price adjustment.

1.3 Steps Involved in Formulating an Econometric Model Although there are of course many different ways to go about the process of model-building, a logical and valid approach would be to follow the steps described in Figure 1.1.

Figure 1.1 Steps involved in formulating an econometric model

The steps involved in the model construction process are now listed and 41

described. Further details on each stage are given in subsequent chapters of this book. Steps 1a and 1b: general statement of the problem This will usually involve the formulation of a theoretical model, or intuition from financial theory that two or more variables should be related to one another in a certain way. The model is unlikely to be able to completely capture every relevant real-world phenomenon, but it should present a sufficiently good approximation that it is useful for the purpose at hand. Step 2: collection of data relevant to the model The data required may be available electronically through a financial information provider, such as Reuters or from published government figures. Alternatively, the required data may be available only via a survey after distributing a set of questionnaires, i.e., primary data. Step 3: choice of estimation method relevant to the model proposed in step 1 For example, is a single equation or multiple equation technique to be used? Step 4: statistical evaluation of the model What assumptions were required to estimate the parameters of the model optimally? Were these assumptions satisfied by the data or the model? Also, does the model adequately describe the data? If the answer is ‘yes’, proceed to step 5; if not, go back to steps 1–3 and either reformulate the model, collect more data, or select a different estimation technique that has less stringent requirements. Step 5: evaluation of the model from a theoretical perspective Are the parameter estimates of the sizes and signs that the theory or intuition from step 1 suggested? If the answer is ‘yes’, proceed to step 6; if not, again return to stages 1–3. Step 6: use of the model When a researcher is finally satisfied with the model, it can then be used for testing the theory specified in step 1, or for formulating forecasts or suggested courses of action. This suggested course of action might be for an individual (e.g., ‘if inflation and GDP rise, buy stocks in sector X’), or as an input to government policy (e.g., ‘when equity markets fall, program trading causes excessive volatility and so should be banned’). It is important to note that the process of building a robust empirical model is an iterative one, and it is certainly not an exact science. Often, the final preferred model could be very different from the one originally proposed, 42

and need not be unique in the sense that another researcher with the same data and the same initial theory could arrive at a different final specification.

1.4 Points to Consider When Reading Articles in Empirical Finance As stated above, one of the defining features of this book relative to others in the area is in its use of published academic research as examples of the use of the various techniques. The papers examined have been chosen for a number of reasons. Above all, they represent (in this author’s opinion) a clear and specific application in finance of the techniques covered in this book. They were also required to be published in a peer-reviewed journal, and hence to be widely available. When I was a student, I used to think that research was a very pure science. Now, having had first-hand experience of research that academics and practitioners do, I know that this is not the case. Researchers often cut corners. They have a tendency to exaggerate the strength of their results, and the importance of their conclusions. They also have a tendency not to bother with tests of the adequacy of their models, and to gloss over or omit altogether any results that do not conform to the point that they wish to make. Therefore, when examining papers from the academic finance literature, it is important to cast a very critical eye over the research – rather like a referee who has been asked to comment on the suitability of a study for a scholarly journal. The questions that are always worth asking oneself when reading a paper are outlined in Box 1.2. BOX 1.2 Points to consider when reading a published paper (1) Does the paper involve the development of a theoretical model or is it merely a technique looking for an application so that the motivation for the whole exercise is poor? (2) Are the data of ‘good quality’? Are they from a reliable source? Is the size of the sample sufficiently large for the model estimation task at hand? (3) Have the techniques been validly applied? Have tests been conducted for possible violations of any assumptions made in the estimation of the model? (4) Have the results been interpreted sensibly? Is the strength of the 43

results exaggerated? Do the results actually obtained relate to the questions posed by the author(s)? Can the results be replicated by other researchers? (5) Are the conclusions drawn appropriate given the results, or has the importance of the results of the paper been overstated? Bear these questions in mind when reading my summaries of the articles used as examples in this book and, if at all possible, seek out and read the entire articles for yourself. This chapter now moves on to cover the fundamental mathematical framework that underpins financial econometrics. This material is intended as a refresher for readers who have covered these topics in the past but require a reminder; students who are seeing these concepts for the first time may find a more thorough treatment covering an entire book useful in addition to this text – see, for example Renshaw (2016) or Swift and Piff (2014), which are both detailed and very accessible.

1.5 Functions 1.5.1 Introduction to Functions The ultimate objective of econometrics is usually to build a model, which may be thought of as a simplified version of the true relationship between two or more variables that can be described by a function. A function is simply a mapping or relationship between an input or set of inputs and an output. We usually write that y, the output, is a function f of x, the input, so y = f (x). f (.) is simply a general method of stating that y is related to x in some fashion. Another way to say this is that f provides a mapping between y and x so that it tells us, for every given value of x, what the corresponding value of y would be. f is a unique (1:1) mapping so that for each value of x there is only one corresponding value of y. The domain of x is defined as the set of values that this variable can take; the range refers to the respective set of values that y can take. Usually, neither the domain nor the range are specified, in which case they can both be assumed to be allowed to take any real values.

1.5.2 Straight Lines y could be a linear function of x, where the relationship can be expressed 44

as a straight line on a graph, or y could be a non-linear function of x, in which case the relationship between the two variables would be represented graphically as a curve. If the relationship is linear, we could write the equation for this straight line as (1.1) y and x are called variables, while a and b are parameters; a is termed the intercept and b is the slope or gradient of the line. The intercept is the point at which the line crosses the y-axis, while the slope measures the steepness of the line. Note that there will be only one value of a and one value of b, although there will be many values of x and of y. a and b could each be any combination of positive, negative or zero. To illustrate, suppose we were trying to model the relationship between a student’s grade-point average y (expressed as a percentage), and the number of hours that they studied throughout the year, x. Suppose further that the relationship can be written as a linear function with y = 25 + 0.05x. Clearly it is unrealistic to assume that the link between grades and hours of study follows a straight line, but let us keep this assumption for now. So the intercept of the line, a, is 25, and the slope, b, is 0.05. What does this equation mean? It means that a student spending no time studying at all (x = 0) could expect to earn a 25% average grade, and for every hour of study time, their average grade should improve by 0.05% – in other words, an extra 100 hours of study through the year would lead to a 5% increase in the grade. Suppose that a particular student wished to score a perfect 100% gradepoint average. How many hours would (s)he need to study? To calculate this, we would need to set y = 100 and then to solve for x: 100 = 25 + 0.05x, so x = 1500 hours. We could construct a table with several values of x and the corresponding value of y as in Table 1.1 and then plot them onto a graph (Figure 1.2). Table 1.1 Sample data on hours of study and grades Hours of study (x)

Grade-point average in % (y)

0

25

100

30

400

45 45

800

65

1000

75

1200

85

Figure 1.2 A plot of hours studied (x) against grade-point average (y)

We can see from the graph that the gradient of this line is positive (i.e., it slopes upwards from left to right). Note that for a straight line, the slope is the same along the whole line; this slope can be calculated from a graph by taking any two points on the line and dividing the change in the value of y by the change in the value of x between the two points. In general, a capital delta, Δ, is used to denote a change in a variable. For example, suppose that we want to take the two points x = 100, y = 30 and x = 1000, y = 75. We could write these two points using a coordinate notation (x,y) and so (100,30) and (1000,75) in this example. We would calculate the slope of the line as (1.2) So indeed, we have confirmed that the slope is 0.05 (although in this case we knew that from the start). Two other examples of straight line graphs are given in Figure 1.3. The gradient of the line can be zero or negative instead of positive. If the gradient is zero, the resulting plot will be a flat (horizontal) straight line. We could then write it as y = 25 + 0x, so that 46

whatever the value of x, y will always be the same (25).

Figure 1.3 Examples of different straight line graphs

If there is a specific change in x, Δx, and we want to calculate the corresponding change in y, we would simply multiply the change in x by the slope, so Δy = bΔ x. As a final point, note that we stated above that the point at which a function crosses the y-axis is termed the intercept. The point at which the function crosses the x-axis is called its root. In the example above, if we take the function y = 25 + 0.05x, set y to zero and rearrange the equation, we would find that the root would be x = −500. In this case, the root of the equation does not have a useful interpretation (as the number of hours studied cannot be negative) but this will not always be the case. The equation for a straight line has one root (except for a horizontal straight line such as y = 4, where there would be no root since it never crosses the x-axis). Further examples of how to calculate the roots of an equation will be given in Section 1.5.3.

1.5.3 Polynomial Functions A linear function is often not sufficiently flexible to be able to accurately describe the relationship between two variables, and so a quadratic function may be used instead. A polynomial simply adds higher order powers of the variable x into the function. In the most general case, we would have an nth order polynomial (a polynomial of order n) (1.3) 47

If n = 2, we have a quadratic equation, if n = 3 a cubic, if n = 4 a quartic and so on. We use polynomials if y depends only on one variable x but in a non-linear way (and so it cannot be expressed as a straight line). An example of the shape of a general polynomial function is given in Figure 1.4.

Figure 1.4 Example of a general polynomial function

Broadly, the higher the order of the polynomial, the more complex will be the relationship between y and x and the more twists and turns there will be in the plot like Figure 1.4. However, usually n = 2, a quadratic equation, is sufficient to describe the function as it seems unlikely that a real series y will rise with x then fall before rising again and so on, which would be the case if it was described by a higher order polynomial. So now we will focus on the quadratic case. We could write the general expression for a quadratic function as (1.4) where x and y are again the variables and a, b, c are the parameters that describe the shape of the function. Note that we have changed notation slightly for simplicity between equations (1.3) and (1.4), writing the slope parameters as b and c rather than b1 and b2. Either notation is equally acceptable so long as we are clear and explain what we mean. A linear function only has two parameters (the intercept, a and the slope, b), but a quadratic has three and hence it is able to adapt to a broader range of relationships between y and x. The linear function is a special case of the quadratic where c is zero. As before, a is the intercept and defines where the function crosses the y-axis; the parameters b and c determine the shape. Quadratic equations can be either ∪-shaped or ∩-shaped. As x becomes 48

very large and positive or very large and negative, the x2 term will dominate the behaviour of y and it is thus c that determines which of these shapes will apply. Figure 1.5 shows two examples of quadratic functions – in the first case c is positive and so the curve is ∪-shaped, while in the second c is negative so the curve is ∩-shaped. We discussed above that the root(s) of an equation is (are) the place(s) where the line crosses the x-axis. Box 1.3 discusses the features of the roots of a quadratic equation and shows how to calculate them.

Figure 1.5 Examples of quadratic functions

BOX 1.3 The roots of a quadratic equation A quadratic equation has two roots The roots may be distinct (i.e., different from one another), or they may be the same (repeated roots); they may be real numbers (e.g., 1.7, −2.357, 4, etc.) or what are known as complex numbers The roots can be obtained either by factorising the equation – i.e., contracting it into parentheses, by ‘completing the square’ or by using the formula (1.5) If b2 > 4ac, the function will have two unique roots and it will cross the x-axis in two separate places; if b2 = 4ac, the function will have two equal roots and it will only cross the x-axis in one 49

place; if b2 < 4ac, the function will have no real roots (only complex roots), it will not cross the x-axis at all and thus the function will always be above the x-axis.

EXAMPLE 1.1 Determine the roots of the following quadratic equations 1. y = x2 + x − 6 2. y = 9x2 + 6x + 1 3. y = x2 − 3x + 1 4. y = x2 −4x

SOLUTION We would solve these equations by setting them in turn to zero. We could then use the quadratic formula from equation (1.5) in each case, although it is usually quicker to determine first whether they factorise (see Box 1.3). 1. x2 + x − 6 = 0 factorises to (x − 2)(x + 3) = 0 and thus the roots are 2 and −3, which are the values of x that set the function to zero. In other words, the function will cross the x-axis at x = 2 and x = −3. 2. 9x2 + 6x + 1 = 0 factorises to (3x + 1)(3x + 1) = 0 and thus the roots are and This is known as repeated roots – since this is a quadratic equation there will always be two roots but in this case they are both the same. We call the expression 9x2 + 6x + 1 a perfect square. Here the plot of y against x would touch, but not cross, the x-axis at 3. x2 − 3x + 1 = 0 does not factorise and so the formula must be used with a = 1, b = −3, c = 1 and the roots are 0.38 and 2.62 to two decimal places. 4. x2 − 4x = 0 factorises to x(x − 4) = 0 and so the roots are 0 and 4. The function crosses the x-axis at the points (0,0) and (4,0). Note that all of these equations have two real roots. If we had an equation such as y = 3x2 − 2x + 4, this would not factorise and would have complex roots since b2 − 4ac < 0 in the quadratic formula. A similar situation is illustrated in the lefthand part of Figure 1.5, which does not cross the x50

axis anywhere.

1.5.4 Powers of Numbers or of Variables A number or variable raised to a power is simply a way of writing repeated multiplication. So, for example, raising x to the power 2 means squaring it (i.e., x2 = x × x); raising it to the power 3 means cubing it (x3 = x × x × x), and so on. The number that we are raising the number or variable to is called the index, so for x3, 3 would be the index. There are a few rules for manipulating powers and their indices given in Box 1.4. BOX 1.4 Manipulating powers and their indices Any number or variable raised to the power one is simply that number or variable, e.g., 31 = 3, x1 = x, and so on. Any number or variable raised to the power zero is one, e.g., 50 = 1, x0 = 1, etc., except that 00 is not defined (i.e., it does not exist). If the index is a negative number, this means that we divide one by that number – for example, If we want to multiply together a given number raised to more than one power, we would add the corresponding indices together – for example, x2 × x3 = x2x3 = x2+3 = x5. The general rule is xa × xb = xa+b. If we want to calculate the power of a variable raised to a power (i.e., the power of a power), we would multiply the indices together – for example, (x2)3 = x2×3 = x6. The general rule is (xa)b = xa×b. If we want to divide a variable raised to a power by the same variable raised to another power, we subtract the second index from the first – for example, The general rule is If we want to divide a variable raised to a power by a different variable raised to the same power, the following result applies

The power of a product is equal to each component raised to that power – for example, (x × y)3 = x3 × y3. It is important to note that the indices for powers do not have to be 51

integers. For example, is the notation we would use for taking the square root of x, sometimes written Other, non-integer powers are also possible, but are harder to calculate by hand (e.g., x0.76, x−0.27, etc.) In general, the nth root of x.

1.5.5 The Exponential Function It is sometimes the case that the relationship between two variables is best described by an exponential function – for example, when a variable y grows (or reduces) at a rate in proportion to its current value x, in which case we would write y = ex · e is a simply number: 2.71828…In fact, e can be derived by letting n in the following expression tend towards infinity (1.6) Alternatively, we can define e as the result from the following infinite sum (1.7) where ! denotes a factorial (e.g., 4! = 4 × 3 × 2 × 1). The exponential function has several useful properties, including that it is its own derivative (see Section 1.6.1 below) and thus the gradient of the function ex at any point is also ex; it is also useful for capturing the increase in value of an amount of money that is subject to compound interest. The exponential function can never be negative, so when x is negative, y is close to zero but positive. It crosses the y-axis at one and the slope increases at an increasing rate from left to right, as shown in Figure 1.6.

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Figure 1.6 A plot of an exponential function

1.5.6 Logarithms Logarithms were invented before computers and pocket calculators were widely available to simplify cumbersome calculations, since exponents can then be added or subtracted, which is easier than multiplying or dividing the original numbers. While logarithmic transformations are no longer necessary for computational ease, they still have important uses in algebra and in data analysis. For the latter, there are at least three reasons why log transforms may be useful. First, taking a logarithm can often help to rescale the data so that their variance is more constant, which overcomes a common statistical problem known as heteroscedasticity, discussed in detail in Chapter 5. Second, logarithmic transforms can help to make a positively skewed distribution closer to a normal distribution. Third, taking logarithms can also be a way to make a non-linear, multiplicative relationship between variables into a linear, additive one. These issues will also be discussed in some detail in Chapter 5. To motivate how logs work, consider the power relationship 23 = 8. Using logarithms, we would write this as log28 = 3, or ‘the log to the base 2 of 8 is 3’. Hence we could say that a logarithm is defined as the power to which the base must be raised to obtain the given number. More generally, if ab = c, then we can also write loga c = b. Natural logarithms, also known as logs to base e, are more commonly used and more useful mathematically than logs to any other base. A log to base e is known as a natural or Napierian logarithm, denoted 53

interchangeably by ln(y) or log(y). Taking a natural logarithm is the inverse of a taking an exponential, so sometimes the exponential function is called the antilog. The log of a number less than one will be negative, e.g., ln(0.5) ≈ −0.69. We cannot take the log of a negative number (so ln(-0.6), for example, does not exist). The properties of logarithmic functions or ‘laws of logs’ describe the way that we can work with logs or manipulate expressions using them. These are presented in Box 1.5. BOX 1.5 The laws of logs For variables x and y ln (x y) = ln (x) + ln (y) ln (x/y) = ln (x) − ln (y) ln (yc) = c ln (y) ln (1) = 0 ln (1/y) = ln (1) − ln (y) = −ln (y). ln(ex) = eln(x) = x

If we plot a log function, y = ln(x), it would cross the x-axis at one, as in Figure 1.7. It can be seen that as x increases, y increases at a slower rate, which is the opposite to an exponential function where y increases at a faster rate as x increases.

Figure 1.7 A plot of a logarithmic function

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1.5.7 Inverse Functions If we have a function such that y = f (x), we would write the inverse as x = f−1(y). To give a simple example of a linear equation, if y = 6x − 3, the inverse function would be a rearrangement of the function to make x the subject: x = (y + 3)/6. For polynomials of order n, there could be up to n possible inverse functions, although the inverse of a function will not always exist.

1.5.8 Sigma Notation If we wish to add together several numbers (or observations from variables), the sigma or summation operator can be very useful. Σ means ‘add up all of the following elements’. For example, Σ(1, 2, 3) = 1 + 2 + 3 = 6. In the context of adding the observations on a variable, it is helpful to add ‘limits’ to the summation (although note that the limits are not always written out if the meaning is obvious without them). So, for instance, we might write

where the i subscript is called an index, 1 is the lower limit and 4 is the upper limit of the sum. This would mean adding all of the values of x from x1 to x4. It might be the case that one or both of the limits is not a specific number – for instance, which would mean x1 + x2 + … + xn, or sometimes we simply write to denote a sum over all the values of the index i. It is also possible to construct a sum of a more complex combination of variables, such as where xi and zi are two separate random variables. It is important to be aware of a few properties of the sigma operator. For example, the sum of the observations on a variable x plus the sum of the observations on another variable z is equivalent to the sum of the observations on x and z first added together individually (1.8) The sum of the observations on a variable x each multiplied by a constant c is equivalent to the constant multiplied by the sum 55

(1.9) But the sum of the products of two variables is not the same as the product of the sums (1.10) We can write the left-hand side (LHS) of equation (1.10) as (1.11) whereas the right-hand side (RHS) of equation (1.10) is written (1.12) We can see that equations (1.11) and (1.12) are different since the latter contains many ‘cross-product’ terms such as x1z2, x3z6, x9z2, etc., whereas the former does not. If we sum n identical elements (i.e., we add a given number to itself n times), we obtain n times that number (1.13) Suppose that we sum all of the n observations on a series, xi – for example, the xi could be the daily returns on a stock (which are not all the same), we would obtain (1.14) So the sum of all of the observations is, from the definition of the mean, equal to the number of observations multiplied by the mean of the series, Notice that the difference between this situation in equation (1.14) and the previous one in equation (1.13) is that now the xi are different from one another whereas before they were all the same (and hence no i subscript 56

was necessary). Finally, note that it is possible to have multiple summations, which can be conducted in any order, so for example

would mean sum over all of the i and j subscripts, but we could either sum over the j’s first for each value of i or sum over the i’s first for each value of j. Usually, the convention is that the inner sum (in this case the one that runs over j from 1 to m would be conducted first – i.e., separately for each value of i).

1.5.9 Pi Notation Similar to the use of sigma to denote sums, the pi operator (П) is used to denote repeated multiplications. For example (1.15) means ‘multiply together all of the xi for each value of i between the lower and upper limits’. It also follows that

For example, the product

is equal to 12 × 22 × 32 × 42 = 1 × 4 × 9 × 16 = 576. Sometimes we need to calculate the geometric mean of a series. If the series contains n elements, this would mean taking the nth root. For example, as we will see in Chapter 2, we would calculate the holding period return on an investment paying a return in each period (assume this is a year) i of ri where there a total of n years as

57

To calculate the average return in each year, we would take the geometric mean (i.e., the nth root) of this, as

and then we would subtract one at the end. A detailed illustration will be given in Section 2.6 of Chapter 2.

1.5.10 Functions of More than one Variable All the examples we have examined so far in this section involve situations where y is a function of a single variable x, but it is also possible for y to be a function of several variables. Returning to the example in Table 1.1 to illustrate, we might suppose that grades (y) depend on hours of study (x1) and hours of tutoring (x2), so we would write (1.16) where a is still interpreted as an intercept, but there are now two slopes: b1 measures how much y varies with changes in x1 while b2 measures how much y varies with changes in x2. In order to plot such a function, we would need a three-dimensional representation. This notation will be very useful in later chapters when we examine relationships between many variables and we can continue to extend the model in exactly the same way according to how many variables we have included.

1.6 Differential Calculus The effect of the rate of change of one variable on the rate of change of another is measured by a mathematical derivative. If the relationship between the two variables can be represented by a curve, the gradient of the curve will be this rate of change. Consider a variable y that is some function f of another variable x, i.e., y = f (x). The derivative of y with respect to x is written

or sometimes written as

58

This term measures the instantaneous rate of change of y with respect to x, or in other words, the impact of an infinitesimally small change in x. Notice the difference between the notations Δy and dy – the former refers to a change in y of any size, whereas the latter refers specifically to an infinitesimally small change.

1.6.1 Differentiation: the Fundamentals The basic rules of differentiation are as follows 1. The derivative of a constant is zero

This is because y = 10 would be represented as a horizontal straight line on a graph of y against x, and therefore the gradient of this function is zero. 2. The derivative of a linear function is simply its slope

But non-linear functions will have different gradients at each point along the curve. In effect, the gradient at each point is equal to the gradient of the tangent at that point – see Figure 1.8. Notice that the gradient will be zero at the point where the curve changes direction from positive to negative or from negative to positive – this is known as a turning point or equivalently as a stationary point.

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Figure 1.8 The tangents to a curve

3. The derivative of a power function n of x

For example

4. The derivative of a power of an entire function such as [f (x)]n is given by

5. The derivative of a sum is equal to the sum of the derivatives of the individual parts. Similarly, the derivative of a difference is equal to the difference of the derivatives of the individual parts

while 60

6. The derivative of the log of x is given by 1/x

7. The derivative of the log of a function of x is the derivative of the function divided by the function

For example, the derivative of ln(x3 + 2x − 1) is given by

8. The derivative of an exponential of x is itself, so if y = ex

More generally, the derivative of a function of an exponential is the derivative of the function multiplied by the exponential of the function, so if y = ef(x)

So, to illustrate, if

1.6.2 Derivatives of Products and Quotients Suppose that we have two functions multiplied together or one function divided by another function (recall that these are known as a product and a quotient, respectively). How would we differentiate these? Fortunately, both are fairly straight-forward. For a product, which could be written as y = f (x)g(x), the rule is

For a quotient, written as

the rule is 61

Let us look at a couple of simple examples. Suppose that we have a product of two functions, y = (3x3 + 7x2)(−2x2 − 6). To differentiate this, product, we can view it as two functions, y = f (x)g(x) and then we simply differentiate the first part, f(x), multiplying that derivative by the second part, g(x), unaltered, and then add the derivative of the second part multiplied by the first part unaltered

Again, it would be possible to simplify this expression but this is left as an exercise. Now, suppose the quotient that we wish to differentiate is

Following the quotient rule, the derivative would be

1.6.3 Higher Order Derivatives It is possible to differentiate a function more than once to calculate the second order, third order, …, nth order derivatives. The notation for the second order derivative (which is usually just termed the second derivative, and which is the highest order derivative that we will need in this book) is

To calculate second order derivatives, we simply differentiate the function with respect to x and then we differentiate it again. For example, suppose that we have the function

The first order derivative is

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The second order derivative is

The second order derivative can be interpreted as the gradient of the gradient of a function – i.e., the rate of change of the gradient. We said above that at the turning point of a function its gradient will be zero. How can we tell, then, whether a particular turning point is a maximum or a minimum? In other words, is the shape of the function for that value of x a ∪ or a ∩? The answer is that to do this we would look at the second derivative. When a function reaches a maximum, its second derivative is negative, while it is positive for a minimum. For example, consider the quadratic function y = 5x2 + 3x − 6. We already know that since the squared term in the equation has a positive sign (i.e., it is 5 rather than, say, −5), the function will have a ∪-shape rather than an ∩-shape, and thus it will have a minimum rather than a maximum. But let us also demonstrate this using differentiation

Since the second derivative is positive, the function indeed has a minimum because the rate of change of the slope is positive as the gradient switches from negative on the left of the minimum to zero at the minimum to positive on the right of the minimum. To find where this minimum is located, take the first derivative, set it to zero and solve it for x. So we have 10x + 3 = 0, and thus If x = −0.3, the corresponding value of y is found by substituting −0.3 into the original function y = 5x2 + 3x − 6 = 5 × (−0.3)2 + (3 × −0.3) − 6 = −6.81. Therefore, the minimum of this function is found at (−0.3, −6.81). What if the second derivative of a function is zero for a particular value of x? In such cases, the function is at a point of inflection. Turning points and points of inflection are both types of stationary point. At a point of inflection, the figure has neither a ∪-shape or a ∩-shape but something more like an ‘S’. To illustrate, consider the function y = (x + 6)3 − 5. Its first derivative is f′′(x) = 3(x + 6)2. The second derivative is f″(x) = 6(x + 6). Suppose that we are interested in evaluating the shape of the function at x = −6. At this point, y = −5, f′(x) = 0 and f″ = 0 so this is a point of inflection. We plot the original function, y = f (x), the first derivative function, y = f′(x), and the 63

second derivative function, y = f″(x), in Figure 1.9.

Figure 1.9 y = f(x), its first derivative and its second derivative around the

point x = −6

1.6.4 Differentiation of Functions of Functions Using the Chain Rule In the section above we saw how to differentiate powers of functions and logarithms of functions. These are just special cases of a more general situation where we might want to differentiate a function of a function, y = f (g(x)). In such situations, we effectively split the process into two parts: we differentiate y with respect to g and then multiply it by the derivative of g with respect to x. We can write this as

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It is easy to see why this approach is often known as the chain rule of differentiation. As an illustration, suppose that we wish to differentiate the function y = (4x3 − 6x + 4)4. In this case we would have g(x) = 4x3 − 6x + 4 and y = g4. The derivative of y with respect to g is

and the derivative of g with respect to x is

Putting these together, the derivative of y with respect to x is:

It may be possible to simplify this function but we leave it in its factorised form. EXAMPLE 1.2: Utility Functions In economics, utility provides a measure of the satisfaction that a consumer derives from a good or service that they have purchased. In finance, the concept is usually used to measure how satisfaction changes with differing levels of (terminal, i.e., end of period) wealth or with risk and return. Utilities constitute a useful illustration of how the concepts of functions and differentiation can be applied in finance. Let us start by considering utility as a function of wealth. We would write the utility function as U = f (W). Many such utility functions would be possible, e.g. 1. U = 5 + 8W 2. U = 30 − e0.5W 3. U = 100W + 0.5W2 4. U = ln(W) But would they all make sense as utility functions and what are the properties that we would want a utility function to have? 65

SOLUTION For a utility function to be plausible, we usually have two requirements. First, we would want to ensure that the investor has a positive marginal utility of wealth – in other words, utility always rises with wealth or mathematically, dU/dW > 0. We would also usually expect that investors are risk averse – i.e., they would reject a fair gamble or they prefer less risk to more. When utility is a function of wealth, it turns out that the condition for the investor to be risk averse is d2U/dW2 < 0. This condition also implies that marginal utility diminishes with wealth – in other words, I get more utility the more wealth I have, but each additional unit of wealth gives me less and less additional satisfaction. This makes intuitive sense. For completeness, note that a risk neutral investor would be indifferent to a gamble and the second derivative of their utility function with respect to wealth would be: d2U/dW2 = 0. A risk loving investor who would prefer more risk to less and who would therefore accept a fair gamble has a second derivative greater than zero: d2U/dW2 > 0. So to evaluate the plausibility of each of the four utility functions above, we would need to differentiate each of them twice and determine whether the first derivative is positive and the second derivative negative. These would be: 1. U = 5 + 8W, dU/DW = 8, d2U/DW2 = 0 2. U = 30 − 30e0.5W, dU/DW = −15e0.5W, d2U/DW2 = −7.5e0.5W 3. U = 100W + 0.5W2, dU/DW = 100 + W, d2U/DW2 = 1 4. U = ln(W), dU/DW = 1/W, d2U/DW2 = −1/W2 Utility function 1 is a linear equation, sloping upwards. The first derivative is positive for all values of W and so the investor having this utility function would have a positive marginal utility of wealth but the second derivative is zero and thus such an investor would be risk neutral. Utility function 2 has a first derivative that is negative (so that the investor prefers less wealth to more) and a second derivative that is also negative for all values of W since eax will be positive for any (positive or negative) value of a and thus the investor would be risk averse. The third utility function has a first derivative that is positive for any value of W greater than −100, and a second derivative that is positive everywhere and thus the investor would be risk loving. Finally, the fourth utility function has a first derivative that is positive 66

for all positive values of W but a negative second derivative for all values of W. So overall, we would conclude that the fourth utility function is the most appropriate of the four to describe a typical investor as it is the only one having the required properties of a positive first derivative and a negative second derivative.

1.6.5 Partial Differentiation In the case where y is a function of more than one variable (e.g., y = f (x1, x2, …, xn)), it may be of interest to determine the effect that changes in each of the individual x variables would have on y. The differentiation of y with respect to only one of the variables, holding the others constant, is known as partial differentiation. The partial derivative of y with respect to a variable x1 is usually denoted

All of the rules for differentiation explained above still apply and there will be one (first order) partial derivative for each variable on the RHS of the equation. We calculate these partial derivatives one at a time, treating all of the other variables as if they were constants. To give an illustration, suppose The partial derivative of y with respect to x1 would be

while the partial derivative of y with respect to x2 would be

As we will see in Chapter 3, the ordinary least squares (OLS) estimator gives formulae for the values of the parameters that minimise the function given by The minimum of L (the residual sum of squares) is found by partially differentiating this function with respect to and then separately with respect to and setting these partial derivatives to zero. Therefore, partial differentiation has a key role in deriving the main approach to parameter estimation that we use in econometrics – see Appendix 3.1 at the end of Chapter 3 for a demonstration of this application.

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1.6.6 Functions that Cannot be Differentiated Fortunately, it is possible to differentiate the majority of functions of interest to us in finance, but are there any formulations where it is not possible to calculate the gradient? The answer is that there are particular difficulties where a function is discontinuous or, in other words, it contains a jump (either up or down). For example, if we have a function y = f (x) which takes a certain form when y is positive or zero and a different form when y is negative such as (1.17) It would not be possible to differentiate this function, which is known as a piecewise linear model, since each of the pieces (≥ 0 and < 0) are linear functions of x but overall it is non-linear. These models will be discussed in more detail in Chapter 10.

1.6.7 Derivatives in Use in Finance What do we actually use differentiation for? A key use relates to the concept of what happens at the margin – in other words, what is the effect of an infinitesimally small change in x on y – this is exactly the interpretation of the slope of a function at a specific value of x. In reality, we usually weaken this slightly to say that the derivative of y with respect to x can be used to measure the effect of a unit change in x on y. This is a very useful concept that is widely used in measuring marginal utility, marginal propensity to save as income changes, etc. – for instance, what is the effect of a one-unit change in wealth upon the utility of an investor? Differentiation relates unit changes in x to unit changes in y but it will often be of interest to consider what happens to y if x changes by one percent rather than one unit. This would be measured by an elasticity. The formula for calculating an elasticity of y with respect to x would be given by (1.18) EXAMPLE 1.3 Suppose that the demand for an on-line stock brokerage account is 68

given by the following function where q is the number of trades made per month and p is the fee charged per trade. If p =£20, calculate the price elasticity of demand. SOLUTION To solve this, we first need to calculate the derivative of q with respect to p, which is very straightforward as it is a linear function: dq/dp = −500. To then calculate the elasticity, we need to calculate the value of q that corresponds to the value of p in the question (20). If p = 20, q = 100,000 − (500 × 20) = 90,000. We then calculate the elasticity as

This would be interpreted as implying that a 1% increase in the fee per trade would reduce the number of trades by 0.111%. Since this figure is less than one in absolute value, we would conclude that demand for brokerage services is inelastic and thus the firm may have the opportunity to increase its revenue and profits by raising prices.

1.6.8 Integration Integration is the opposite of differentiation, so that if we integrate a function and then differentiate the result, we get back the original function. Recall that derivatives give functions for calculating the slope of a curve; integration, on the other hand, is used to calculate the area under a curve (between two specific points). Further details on the rules for integration are beyond the scope of this book since the mathematical technique is not needed for any of the econometric approaches we will employ, but it will be useful to be familiar with the general concept. For further reading, see for example Renshaw (2016, Chapter 18).

1.7 Matrices Before we can work with matrices, we need to define some terminology and to distinguish between a scalar, a vector and a matrix. A scalar is simply a single number (although it need not be a whole number – e.g., 3, −5, 0.5 are all scalars) A vector is a one-dimensional array of numbers (see below for 69

examples) A matrix is a two-dimensional collection or array of numbers. The size of a matrix is given by its numbers of rows and columns. Matrices are very useful and important ways for organising sets of data together, which make manipulating and transforming them much easier than it would be to work with each constituent of the matrix separately. Matrices are widely used in econometrics and finance for solving systems of linear equations, for deriving key results and for expressing formulae in a succinct way. Sometimes bold-faced type is used to denote a vector or matrix (e.g., A), although in this book we will not do so – hopefully it should be obvious whether an object is a scalar, vector or matrix from the context, or this will be clearly stated. Some useful features of matrices and explanations of how to work with them are now described. The dimensions of a matrix are quoted as R × C, which is the number of rows by the number of columns. Each element in a matrix is referred to using subscripts. For example, suppose a matrix M has two rows and four columns. The element in the second row and the third column of this matrix would be denoted m23, so that more generally mij refers to the element in the ith row and the jth column. Thus a 2 × 4 matrix would have elements

Vectors are special cases of matrices where there is only one column or only one row. If a matrix has only one row, it is known as a row vector, which will be of dimension 1 × C, where C is the number of columns A matrix having only one column is known as a column vector, which will be of dimension R × 1, where R is the number of rows

When the number of rows and columns is equal (i.e., R = C), it would be said that the matrix is square, as is the following 2 × 2 matrix 70

A matrix in which all the elements are zero is known as a zero matrix

A symmetric matrix is a special type of square matrix that is symmetric about the leading diagonal (the diagonal line running through the matrix from the top left to the bottom right), so that mij = mji ∀ i, j

A diagonal matrix is a square matrix which has non-zero terms on the leading diagonal and zeros everywhere else

A diagonal matrix with 1 in all places on the leading diagonal and zero everywhere else is known as the identity matrix, denoted by I. By definition, an identity matrix must be symmetric (and therefore also square)

The identity matrix is essentially the matrix equivalent of the number one. Multiplying any matrix by the identity matrix of the appropriate size results in the original matrix being left unchanged. So for any matrix M

1.7.1 Operations with Matrices 71

In order to perform operations with matrices (e.g., addition, subtraction or multiplication), the matrices concerned must be conformable. The dimensions of matrices required for them to be conformable depend on the operation. Addition and subtraction of matrices requires the matrices concerned to be of the same order (i.e., to have the same number of rows and the same number of columns as one another). The operations are then performed element by element

Multiplying or dividing a matrix by a scalar (that is, a single number), implies that every element of the matrix is multiplied by that number

More generally, for two matrices A and B of the same order and for c a scalar, the following results hold

Multiplying two matrices together requires the number of columns of the first matrix to be equal to the number of rows of the second matrix. Note also that the ordering of the matrices is important when multiplying them, so that in general, AB ≠ BA. When matrices are multiplied together, the resulting matrix will be of size (number of rows of first matrix × number of columns of second matrix), e.g., if we multiply a (3 × 2) matrix by a (2 × 4) matrix, the result is a (3 × 4) matrix: (3 × 2) × (2 × 4) = (3 × 4). In terms of determining the dimensions of the matrix, it is as if the number of columns of the first 72

matrix and the number of rows of the second cancel out.1 This rule also follows more generally, so that (a × b) × (b × c) × (c × d) × (d × e) = (a × e), etc. The actual multiplication of the elements of the two matrices is done by multiplying along the rows of the first matrix and down the columns of the second

In general, matrices cannot be divided by one another. Instead, we achieve the same sort of outcome by multiplying by the inverse – see below. The transpose of a matrix, written A′ or AT, is the matrix obtained by transposing (switching) the rows and columns of a matrix

If A is of dimensions R × C, A′ will be C × R.

1.7.2 The Rank of a Matrix The rank of a matrix A is given by the maximum number of linearly independent rows (or columns) contained in the matrix. For example,

since both rows and columns are (linearly) independent of one another, but

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as the second column is not independent of the first (the second column is simply twice the first and also the second row is two thirds of the first). A matrix with a rank equal to its dimension, as in the first of these two cases, is known as a matrix of full rank. A matrix that is less than of full rank is known as a short rank matrix, and such a matrix is also termed singular. Three important results concerning the rank of a matrix are Rank(A) = Rank(A′) Rank(AB) ≤ min(Rank(A), Rank(B)) Rank(A′A) = Rank(AA′) = Rank(A)

1.7.3 The Inverse of a Matrix The inverse of a matrix A, where defined, is denoted A−1. It is that matrix which, when pre-multiplied or post-multiplied by A, will result in the identity matrix

The inverse of a matrix exists only when the matrix is square and nonsingular – that is, when it is of full rank. The inverse of a 2 × 2 nonsingular matrix whose elements are

will be given by

The expression in the denominator above to the left of the matrix (ad − bc) is the determinant of the matrix, and will be a scalar. If this determinant is zero, the matrix is singular, and thus not of full rank so that its inverse does not exist. For example, if

ad − bc = 12 − 12 = 0 so this matrix is singular since the second column is six times the first (or looking at it another way, the second row is double the first). We usually write the determinant of a matrix using |.| (the same notation as for the absolute value of a variable). So |A| is the determinant of matrix A. 74

EXAMPLE 1.4 If the matrix is

the inverse will be

As a check, multiply the two matrices together and it should give the identity matrix – the matrix equivalent of one (analogous to )

= I, as required. The calculation of the inverse of an N × N matrix for N > 2 is more complex. Two of the most common approaches to finding the inverse of a larger matrix are known as the method of determinants and the GaussJordan elimination method. These are beyond the scope of this text but see Wisniewski (2013), for example, for further details. Properties of the inverse of a matrix include I−1 = I (A−1)−1 = A (A′)−1 = (A−1)′ (AB)−1 = B−1A−1

1.7.4 The Trace of a Matrix The trace of a square matrix is the sum of the terms on its leading diagonal. For example, the trace of the matrix

written Tr(A), is 3 + 9 = 12. Some important properties of the trace of a matrix are

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Tr(cA) = cTr(A) Tr(A′) = Tr(A) Tr(A + B) = Tr(A) + Tr(B) Tr(IN) = N

1.7.5 The Eigenvalues of a Matrix The concept of the eigenvalues of a matrix is necessary for testing for long-run relationships between series using what is known as the Johansen cointegration test used in Chapter 8. Let П denote a p × p square matrix, c denote a p × 1 non-zero vector, and λ denote a set of scalars. λ is called a characteristic root or set of roots of the matrix if it is possible to write

This equation can also be written as

where Ip is an identity matrix, and hence

Since c ≠ 0 by definition, then for this system to have a non-zero solution, the matrix (П − λIp) is required to be singular (i.e., to have a zero determinant)

For example, let П be the 2 × 2 matrix

Then the characteristic equation is

This gives the solutions λ = 6 and λ = 3. The characteristic roots are also known as eigenvalues. The eigenvectors would be the values of c 76

corresponding to the eigenvalues. Some properties of the eigenvalues of any square matrix A are the sum of the eigenvalues is the trace of the matrix the product of the eigenvalues is the determinant the number of non-zero eigenvalues is the rank For a further illustration of the last of these properties, consider the matrix

Its characteristic equation is

which implies that

This determinant can also be written (0.5 − λ)(0.35 − λ) − (0.7 × 0.25) = 0 or

or which can be factorised to λ (λ − 0.85) = 0. The characteristic roots are therefore 0 and 0.85. Since one of these eigenvalues is zero, it is obvious that the matrix П cannot be of full rank. In fact, this is also obvious from just looking at П, since the second column is exactly half the first. KEY CONCEPTS The key terms to be able to define and explain from this chapter are functions turning points the chain rule powers 77

exponentials sigma and pi notation quadratic equations inverse of a matrix eigenvalues roots derivatives and differentiation products and quotients indices polynomials logarithms conformable matrix rank of a matrix eigenvectors

SELF-STUDY QUESTIONS 1. (a) If f (x) = 3x2 − 4x + 2, find f (0), f (2), f (−1) (b) If f (x) = 4x2 + 2x − 3, find f (0), f (3), f (a), f (3 + a) (c) Considering your answers to the previous question part, in general does f (a) + f (b) = f (a + b)? Explain. 2. Simplify the following as much as possible (a) 4x5 × 6x3 (b) 3x2 × 4y2 × 8x4 × −2y4 (c) (4p2q3)3 (d) 6x5 ÷ 3x2 (e) 7y2 ÷ 2y5 (f) (g) (xy)3 ÷ x3y3 (h) (xy)3 − x3y3 3. Solve the following (a) 1251/3 (b) 641/3 78

(c) 161/4 (d) 93/2 (e) 92/3 (f) 811/2 + 641/2 + 641/3 4. Write each of the following as a prime number raised to a power (a) 9 (b) 625 (c) 125−1 5. Solve the following equations (a) 3x − 6 = 6x − 12 (b) 2x − 304x + 8 = x + 9 − 3x + 4 (c) 6. Write out all of the terms in the following and evaluate them (a) (b) (c) (d)

with n = 4 and x = 3 with x = 2

(e) 7. Write the equations for each of the following lines (a) Gradient = 3, intercept = −1 (b) Gradient = −2, intercept = 4 (c) crosses y-axis at 3 (d) crosses x-axis at 3 (e) Intercept 2 and passing through (3,1) (f) Gradient 4 and passing through (−2,−2) (g) Passes through x = 4, y = 2 and x = −2, y = 6 8. Differentiate the following functions twice with respect to x (a) y = 6x (b) y = 3x2 + 2 (c) y = 4x3 + 10 (d) 79

(e) y = x (f) y = 7 (g) (h) y = 3 ln x (i) y = ln(3x2) (j) 9. Differentiate the following functions partially with respect to x and (separately) partially with respect to y (a) z = 10x3 + 6y2 − 7y (b) z = 10xy2 − 6 (c) z = 6x (d) z = 4 10. Factorise the following expressions (a) x2 − 7x − 8 (b) 5x − 2x2 (c) 2x2 − x − 3 (d) 6 + 5x − 4x2 (e) 54 − 15x − 25x2 11. Express the following in logarithmic form (a) 53 = 125 (b) 112 = 121 (c) 64 = 1296 12. Evaluate the following (without using a calculator) (a) ln10 10,000 (b) ln2 16 (c) ln10 0.01 (d) ln5 125 (e) lne e2 13. Express the following logarithms using powers (a) ln5 3125 = 5 80

(b) (c) ln0.5 8 = −3 14. Write the following as simply as possible as sums of logs of prime numbers (a) ln 60 (b) ln 300 15. Simplify the following as far as possible (a) ln 27 − ln 9 + ln 81 (b) ln 8 − ln 4 + ln 32 16. Solve the following (a) ln x4 − ln x3 = ln 5x − ln 2x (b) ln(x − 1) + ln(x + 1) = 2 ln(x + 2) (c) log10 x = 4 17. Use the result that ln(8) is approximately 2.1 to estimate the following (without using a calculator): (a) ln(16) (b) ln(64) (c) ln(4) 18. Solve the following using logs and a calculator (a) 4x = 6 (b) 42x = 3 (c) 32x−1 = 8 19. Find the minima of the following functions. In each case, state the value of the function at the minimum (a) y = 6x2 − 10x − 8 (b) y = (6x2 − 8)2 20. Construct an example not used elsewhere in this book to demonstrate that for two conformable matrices A and B, (AB)−1 = B−1A−1. 21. Suppose that we have the following four matrices

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(a) Which pairs of matrices can be validly multiplied together? For these pairs, perform the multiplications. (b) Calculate 2A, 3B, (c) Calculate Tr(A), Tr(B), Tr(A + B) and verify that Tr(A) + Tr(B) = Tr(A + B) (d) What is the rank of the matrix A? (e) Find the eigenvalues of the matrix (A + B) (f) What will be the trace of the identity matrix of order 12? 22. (a) Add

(b) Subtract

(c) Calculate the inverse of

(d) Does the inverse of the following matrix exist? Explain your answer

23. A researcher suggests that the US dollar – British pound exchange rate is a function of US and UK interest rates. (a) Write an equation for this function (b) What signs would we expect for the parameters in this function and why? (c) Give an example of parameter values that would lead the US interest rate to have three times the effect on the exchange rate as the UK interest rate 24. Give an example of a function that cannot be differentiated and explain why.

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1

Of course, the actual elements of the matrices themselves do not cancel out – this is just a simple rule of thumb for calculating the dimensions of the matrix resulting from a multiplication.

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2 Statistical Foundations and Dealing with Data

LEARNING OUTCOMES In this chapter, you will learn how to Construct minimum variance and mean-variance efficient portfolios Compute summary statistics for a data series Manipulate expressions using the expectations, variance and covariance operators Compare nominal and real series Deflate series to allow for inflation Distinguish between different types of data Compound and discount cashflows Calculate present values and future values Use standard formulae to value stocks and bonds Calculate asset price returns

This chapter covers the statistical building blocks that are essential for a good understanding of the rest of the book and provides an introduction to random variables and to dealing with and summarising financial data. It also explains how to work with discounted cashflows, how to calculate present values and how to compute returns in both nominal and real terms using both discrete and continuous compounding.

2.1 Probability and Probability Distributions This section discusses and presents the theoretical expressions for the 84

mean and variance of a random variable. A random variable is one that can take on any value from a given set and where this value is determined at least in part by chance. By their very nature, random variables are not perfectly predictable. Most data series in economics and finance are best considered as random variables, although there might be some measurable structure underlying them as well so they are not purely random. It is often helpful to think of such series as being made up of a fixed part (which we can model and forecast) and a purely random part, which we cannot forecast. The data that we use in building econometric models either come from experiments or, more commonly, are observed in the ‘real world’. The outcomes from an experiment can often only take on certain specific values – i.e., they are discrete random variables. For example, the sum of the scores from throwing two dice could only be a number between two (if we throw two ones) and twelve (if we throw two sixes). We could calculate the probability of each possible sum occurring and plot it on a diagram, such as Figure 2.1. This would be known as a probability distribution function, which shows the various outcomes that are possible and how likely each one is to occur.

Figure 2.1 The probability distribution function for the sum of two dice

A probability is defined as the likelihood of a particular event happening. For example, we could calculate the probability that it will rain 85

tomorrow, or the probability of scoring a total of seven when we throw two dice. All probabilities must lie between zero and one, with a probability of zero indicating an impossibility and one indicating a certainty. Notice that the sum of the probabilities in Figure 2.1 is, as always, one. Most of the time in finance we work with continuous rather than discrete variables, in which case the plot above would be probability density function (pdf) rather than a distribution function. A continuous random variable can take any value (possibly only within a given range). For example, the amount of time a swimmer takes to complete one length of a pool or the return on a stock index. The time that the swimmer takes could be any positive value, depending on how fast they are! The return on a stock index could take any value greater than –100% – in other words, the most that an investor in the stock can lose is their entire investment (−100%), but there is no maximum to the amount that they can gain. At least in theory, the price could double, quadruple, quintuple, etc. In both cases, the value that the process takes can be defined to any arbitrary level of precision - e.g., the swimmer completing the length in 31 seconds, 31.2 seconds, 31.17 seconds, and so on. Hence, note that for a continuous random variable, the probability that it is exactly equal to a particular number is always zero by definition because the variable could take on any value. There are many continuous distribution (density) functions. The simplest is the uniform distribution, where all of the possible outcomes are equally likely to occur. In this case, the pdf is a horizontal straight line. While conceptually simple, the uniform distribution is not very useful as it describes few variables of interest in economics and finance. The distribution most commonly used to characterise a random variable is a normal or Gaussian (these terms are equivalent) distribution. The normal distribution is easy to work with since it is symmetric, it is unimodal (i.e., only has one peak) and the only pieces of information required to completely specify the distribution are its mean and variance, as discussed in Chapter 5. The normal distribution is particularly useful because many naturally occurring series follow it – for example, the heights, weights and IQ-levels of people in a given sample will in general roughly follow a normal distribution. The normal distribution also has several useful mathematical properties. For example, any linear transformation of a normally distributed random variable will still be normally distributed. So, if y ~ N(μ, σ2), that is, y is 86

normally distributed with mean μ and variance σ2, then a + by ~ N(a + bμ, b2σ2) where a and b are scalars. Furthermore, any linear combination of independent normally distributed random variables is itself normally distributed. Suppose that we have a normally distributed random variable with mean μ and variance σ2. Its probability density function is given by f (y) in the following expression (2.1) Entering values of y into this expression would trace out the familiar ‘bellshape’ of the normal distribution described in Figure 2.2. Notice that if a random variable follows a normal distribution, outcomes close to the mean of the series are more likely to occur than those in the extremes, as represented by the peak of the distribution being in the centre and its height declining further away from the mean. The x-axis on the left side and right side are both asymptotes to the normal distribution – in other words, it gradually gets closer and closer to the x-axis the further away from its mean the value of x becomes.

Figure 2.2 The pdf for a normal distribution

The area under a pdf measures the probabilities, and since the sum of the probabilities of all events occurring is one, the area under a pdf will always be one. Note that for a continuous random variable, we can only talk of the probability that it will take on values within a range (e.g., the 87

probability that y will be between 1 and 2) and not the probability that y will be equal to some number (e.g., 2). Remember from above that the probability of y being an exact number is zero, since y can take any value to an arbitrary degree of precision (e.g., 2.0001, 2.0000001, etc.). A standard normally distributed random variable can be obtained from any normal distribution by subtracting the mean and dividing by the standard deviation (the square root of the variance). A standard normally distributed random variable, usually denoted by Z, would then be written as (2.2) It is usually easier to work with the normal distribution in its standardised form, and only this normal distribution is tabulated (since there are an infinite number of normal distributions with different means and variances, we could not tabulate them all!). Distributions are important in statistics because of their link with probabilities. If we know (or we can assume) the particular distribution that a series follows, then we can calculate the likelihood (probability) that values this series takes will fall within a certain range. For example, if we can assume that a series, y, follows a standard normal distribution, we can calculate the probability that it will take a value of +3 or more. This information can be calculated from the cumulative density function, also sometimes known as the cumulative distribution function, (cdf), which is written F(y). The cdf for a normally distributed random variable has a sigmoid shape, as in Figure 2.3.

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Figure 2.3 The cdf for a normal distribution

More specifically, we can use the cdf to calculate the probability that the random variable lies within a certain range – e.g., what is the probability that y lies between 0.2 and 0.3? This is equivalent to asking what is the area under the normal distribution pdf between 0.2 and 0.3? To obtain this, we would plug y = 0.2 and then separately y = 0.3 into the equation for the cdf and calculate the corresponding value of f (y) in each case. Then the difference between these two values of f (y) would give us the answer. More often, rather than wanting to determine the probability that a random variable lies within a range, we instead want to know the probability that the variable is below a certain value (or above a certain value). So, for example, what is the probability that y is less than 0.4? Effectively, we want to know the probability that y lies between −∞ and 0.4. Thus the probability that y is less than (or equal to) some specific value of y, y0, is equal to the cdf of y evaluated where y = y0 (2.3) Note that there are also alternative versions of the normal distribution table that present the information the other way around. So they show values of Zα and the corresponding values of α – i.e., for a given value of Z, say 1.5, they show the probability of a standard normally distributed random variable being bigger than this rather than less than as in equation (2.3). Table A2.2 in Appendix 2 at the back of this book presents what are known as the critical values for the normal distribution. Effectively, if we plotted the values on the first row, α, against the values in the second row, Zα, then we would trace out the cdf. Looking at the table, if α = 0.1, Zα = 1.2816. So 10% (0.1 in proportion terms) of the normal distribution lies to the right of 1.2816. In other words, the probability that a standard normal random variable takes a value greater than 1.2816 is 10%. Similarly, the probability that it takes a value greater than 3.0902 is 0.1% (i.e., 0.001). We know that the standard normal distribution is symmetric about zero so if P(Z ≥ 1.2816) = 0.1, P(Z ≤ − 1.2816) = 0.1 as well.

2.1.1 The Central Limit Theorem If a random sample of size N : y1, y2, y3, …, yN is drawn from a population 89

that is normally distributed with mean μ and variance σ2, the sample mean, is also normally distributed with mean μ and variance σ2/N. In fact, an important rule in statistics known as the central limit theorem states that the sampling distribution of the mean of any random sample of observations will tend towards the normal distribution with mean equal to the population mean, μ, as the sample size tends to infinity. This theorem is a very powerful result because it states that the sample mean, will follow a normal distribution even if the original observations (y1, y2, …, yN) did not. This means that we can use the normal distribution as a kind of benchmark when testing hypotheses, as discussed more fully in Chapter 3.

2.1.2 Other Statistical Distributions There are many statistical distributions, including the binomial, Poisson, log normal, normal, exponential, t, chi-squared and F, and each has its own characteristic pdf. Different kinds of random variables will be best modelled with different distributions. Many of the statistical distributions are also related to one another, and most (except the normal) have one or more degrees of freedom parameters that determine the location and shape of the distribution. For example, the chi-squared (denoted χ2) distribution can be obtained by taking the sum of the squares of independent normally distributed random variables. If we sum n independent squared normals, the result will be a χ2 with n degrees of freedom. Since it comprises the sum of squares, the chi-squared distribution can only take positive values. Unlike the normal distribution, the chi-squared is not symmetric about its mean value. The F-distribution, which has two degrees of freedom parameters, is the ratio of independent chi-squared distributions, each divided by their degrees of freedom. Suppose that y1 ~ χ2(n1) and y2 ~ χ2(n2) are two independent chi-squared distributions with n1 and n2 degrees of freedom, respectively. Then the ratio will follow an F distribution with (n1, n2) degrees of freedom,

The final, and arguably most important, distribution used in econometrics is the t-distribution. The normal distribution is a special case of the t. The t-distribution can also be obtained by taking a standard normally distributed random variable, Z, and dividing it by the square root of an 90

independent chi-squared distributed random variable (suppose that the latter is called y1), itself divided by its degrees of freedom, n1

The t-distribution is symmetric about zero and looks similar to the normal distribution except that it is flatter and wider. What do we use statistical distributions for? The normal, F, t and chisquared distributions are all used predominantly to make inferences from the sample to the population. This implies making statements about the likely values of the corresponding unobservable population values from the sample values that we have. These ideas will be discussed in considerable detail in Chapter 3 onwards.

2.2 A Note on Bayesian versus Classical Statistics The philosophical approach to model-building adopted in this entire book, as with the majority of others, is that of ‘classical statistics’. Under the classical approach, the researcher postulates a theory and estimates a model to test that theory. Tests of the theory are conducted using the estimated model within the ‘classical’ hypothesis testing framework developed in Chapters 2 to 5. Based on the empirical results, the theory is either refuted or upheld by the data. There is, however, an entirely different approach available for model construction, estimation and inference, known as Bayesian statistics. Under a Bayesian approach, the theory and empirical model work more closely together. The researcher would start with an assessment of the existing state of knowledge or beliefs, formulated into a set of probabilities. These prior inputs, or priors, would be combined with the observed data via a likelihood function. The beliefs and the probabilities would then be updated as a result of the model estimation, resulting in a set of posterior probabilities. Probabilities are thus updated sequentially, as more data become available. The central mechanism, at the most basic level, for combining the priors with the likelihood function, is known as Bayes’ theorem. The Bayesian approach to estimation and inference has found a number of important recent applications in financial econometrics, in particular in the context of volatility modelling (see Bauwens and Laurent, 2002, or Vrontos et al., 2000 and the references therein for some examples), asset 91

allocation (see, for example, Handa and Tiwari, 2006), and portfolio performance evaluation (Baks et al., 2001). The Bayesian setup is an intuitively appealing one, although the resulting mathematics is somewhat complex. Many classical statisticians are unhappy with the Bayesian notion of prior probabilities that are set partially according to judgement. Thus, if the researcher set very strong priors, an awful lot of evidence against them would be required for the notion to be refuted. Contrast this with the classical case, where the data are usually permitted to freely determine whether a theory is upheld or refuted, irrespective of the researcher’s judgement.

2.3 Descriptive Statistics When analysing a series containing many observations, it is useful to be able to describe its most important characteristics using a small number of summary measures. This section discusses the quantities that are most commonly used to describe financial and economic series, known as summary statistics or descriptive statistics. Descriptive statistics are calculated from a sample of data rather than assigned based on theory. Before discussing the most important summary statistics used in work with finance data, we define the terms population and sample, which have precise meanings in statistics, in Box 2.1. BOX 2.1 The population and the sample The population is the total collection of all objects to be studied. For example, in the context of determining the relationship between risk and return for UK stocks, the population of interest would be all time-series observations on all stocks traded on the London Stock Exchange (LSE). The population may be either finite or infinite, while a sample is a selection of just some items from the population. A population is finite if it contains a fixed number of elements. In general, either all of the observations for the entire population will not be available, or they may be so many in number that it is infeasible to work with them, in which case a sample of data is taken for analysis. The sample is usually random, and it should be representative of the population of interest. A random sample is one in which each 92

individual item in the population is equally likely to be drawn. A stratified sample is obtained when the population is split into layers or strata and the number of observations in each layer of the sample is set to try to match the corresponding number of elements in those layers of the population. The size of the sample is the number of observations that are available, or that the researcher decides to use, in estimating the parameters of the model.

2.3.1 Measures of Central Tendency The average value of a series is sometimes known as its measure of location or measure of central tendency. The average is usually thought to measure the ‘typical’ value of a series. There are a number of methods that can be used for calculating averages. The most well known of these is the arithmetic mean (usually just termed ‘the mean’), denoted for a series ri of length N, which is simply calculated as the sum of all values in the series divided by the number of values (2.4) EXAMPLE 2.1 Calculate the mean of the following numbers: 2, 4, −6, 7, 1, 0, 20. The series has N = 7 items. The mean, (to two decimal places).

The two other methods for calculating the average of a series are the mode and the median. The mode measures the most frequently occurring value in a series, which is sometimes regarded as a more representative measure of the average than the mean. Finally, the median is the middle value in a series when the elements are arranged in an ascending order.1 If there is an even number of values in a series, then strictly there are two medians. For example, consider a variable that has taken the values listed in order: {3, 7, 11, 15, 22, 24}, the medians are 11 and 15. Sometimes we take the mean 93

of the two medians, so that the median would be (11 + 15)/2 = 13. Each of these three measures of average has its relative merits and demerits, which will now be discussed. The mean is the most familiar method to most researchers, is most easily used in algebraic formulae (see the discussion on expected values below), and has desirable econometric properties (most notably, it is, under some assumptions, unbiased and efficient as we will demonstrate in Chapter 3). But it can be unduly affected by extreme values (what are often termed outliers) and, in such cases, it may not be representative of most of the data. It should be evident for the example above that the mean of the series, 10.86, is larger than all but one of the data points. If the final data point had been −2 instead of 20, the mean would have been reduced to 7.71, and so just one data point can have a profound effect on the mean of a series. The mode is arguably the easiest to obtain, but is not suitable for continuous, non-integer data (e.g., returns or yields) or for distributions that incorporate two or more peaks (known as bimodal and multi-modal distributions, respectively). The mode has the advantage that, unlike the other two measures of average, the mode is guaranteed to be one of the observations. A commonly presented example of why the mode can be useful is that of a shoemaker who needs to know the number of pairs of shoes of each size to produce and asks his or her apprentice to give him or her one number that summarises the sizes of people’s feet. The mean would be useless in this case, for what use is it to know that the mean shoe size is 8.9? On the other hand, if we know that the modal size is 7, this at least tells us that 7 is a more commonly occurring shoe size than any other. In different situations, especially if the distribution of the variable of interest is skewed (see below), the mode would be less useful. For example, if we were interested in knowing how much money the ‘average’ student gives to charity per month, it would not give us much information to know that the mode is zero. The median is often considered to be a useful representation of the ‘typical’ value of a series, and is robust to outliers, which is valuable if these are not of interest. But the median has the drawback that its calculation is based essentially on one observation. Thus if, say, we had a series containing ten observations and we were to double the values of the top three data points, the median would be unchanged. For example, the median of the set of data points: {1, 1, 1, 1, 100, 94

100, 100} is 1 and the median of {1, 1, 1, 1, 200, 200, 200} is also 1. The Geometric Mean There also exists another method that can be used to estimate the average of a series, known as the geometric mean. As briefly mentioned in Chapter 1, it involves calculating the Nth root of the product of N numbers. In other words, if we want to find the geometric mean of six numbers, we multiply them together and take the sixth root (i.e., raise the product to the power of ). In finance, we usually deal with returns or percentage changes (which could be positive, negative or zero) rather than prices or actual values, and the method for calculating the geometric mean described in the previous paragraph cannot handle zero or negative numbers. Therefore, we use a slightly different approach in such cases. To calculate the geometric mean of a set of N returns,2 we express them as proportions (i.e., on a (−1, 1) scale) rather than percentages (on a (−100, 100) scale), and we would use the formula (2.5) where r1, r2, …, rN are the returns on a single asset or portfolio at each of N points in time and is the calculated value of the geometric mean. Hence what we would do would be to add one to each return, then multiply the resulting expressions together, raise this product to the power 1/N and then subtract one right at the end. Return calculations will be discussed in considerable detail in Section 2.6 later in this chapter. So which method for calculating mean returns (arithmetic or geometric) should we use? The answer is, as usual, that ‘it depends’. Geometric returns give the fixed return on the asset or portfolio that would have been required to match the actual performance, which is not the case for the arithmetic mean. Thus, if you assumed that the arithmetic mean return had been earned on the asset every year, you would not reach the correct value of the asset or portfolio at the end. The reason is that the effect of compounding: the money that you have available to invest in year two will be the sum of the original investment and however much money was made or lost in year one, which implies that the investment values in each year are not independent of one another (even if the individual annual returns are). So if you invested £1000 at time zero, but the fund performed poorly and lost 20% in year one, you would only have £800 going into year two 95

and would need a 25% positive return that year just to get back your original investment. The arithmetic averaging implicitly ignores this compounding effect and assumes that you always had the original investment amount at the start of each new year. Note that if the individual annual returns are already continuously compounded (i.e., log returns – see Box 2.3 on p. 78), then it would be more appropriate to use the arithmetic average to calculate overall performance rather than the geometric average, since with log returns the effect of compounding has already been taken into account. An extensive discussion of compounding and its effects will be presented in Section 2.6 later in this chapter. In fact, the formula that links the simple return, Rt, with the continuously compounded return, rt, is simply (2.6) If we plug some numbers into this equation, we can see that the continuously compounded return will be slightly smaller or more negative than the simple return, and the difference between the two will be bigger for large returns. For example, if the continuously compounded return rt is 2, the equivalent simple return Rt will be 2.02; if rt is 10, Rt will be 10.52, if rt = −4, Rt = −3.92; rt = −20, Rt = −18.13, etc. But it can also be shown that the geometric return is always less than or equal to the arithmetic return, and so the geometric return is a downwardbiased predictor of future performance. Hence, if the objective is to summarise historical performance, the geometric mean is more appropriate, but if we want to forecast future returns, the arithmetic mean is the one to use. Finally, it is worth noting that the geometric mean is evidently less intuitive and less commonly used than the arithmetic mean, but it is less affected by extreme outliers than the latter. There is an approximate relationship which holds between the arithmetic and geometric means, calculated using the same set of returns (2.7) where and are the geometric and arithmetic means respectively and σ2 is the variance of the returns. We can see from this formula that the arithmetic mean is higher than the geometric mean unless there is zero volatility and thus it is hardly surprising that it is more common for fund managers to report their arithmetic mean returns! We can also see that the 96

higher the volatility, the greater will be the difference between the two measures of average returns, and thus the more the arithmetic average will overstate the investor experience and how much his or her money would have grown over time.

2.3.2 Measures of Spread Usually, the average value of a series will be insufficient to adequately characterise a data series, since two series may have the same mean but very different profiles because the observations on one of the series may be much more widely spread about the mean than the other. Hence, another important feature of a series is how dispersed its values are. In finance theory, for example, the more widely spread are the returns around their mean value, the more risky the asset is usually considered to be. Percentiles of a Distribution The percentiles (sometimes also known as the quantiles) of a distribution provide information on where a particular observed value sits within the ordered set of all values. To illustrate, it is common to examine information on the weight of a baby compared with the weights of all other babies, and a parent might be given information that their baby is at the 80th percentile of the distribution of weights. This would imply that this baby was heavier than 80% of all other babies in the database. Or in other words, if we ordered all the babies in a line according to their weight with the lightest on the left and the heaviest on the right, this baby would have 80% of the others to his or her left. On the other hand, if the baby’s weight is at the fifth percentile, this would mean that he/she was heavier than only 5% of other babies (or put equivalently, that 95% of other babies were heavier than him or her). It should already be obvious that, by definition, the median is the 50th percentile, but providing information on other percentiles of an empirical distribution of real data can give us useful clues about its shape. The 0th percentile and the 100th percentile would, respectively, define the minimum and maximum values in the dataset, while the first and fifth percentiles have specific uses in financial risk management, as it is often the case that we want to focus on the lowest 1% or 5% of historical returns that have occurred. The difference between two percentiles can be used as a measure of the spread of a distribution. The simplest such measure is arguably the range, 97

which is calculated by subtracting the smallest observation from the largest. While the range has some uses, it is fatally flawed as a measure of dispersion by its extreme sensitivity to an outlying observation since it is effectively based only on the very lowest and very highest values in a series, and it ignores all of the other data points. A more reliable measure of spread, although it is not widely employed by quantitative analysts, is the semi-interquartile range, sometimes known as the quartile deviation. Calculating this measure involves first ordering the data and then splitting the sample into four parts (quartiles) with equal numbers of observations.3 The second quartile will be exactly at the half way point, and is the median, as described above. But the interquartile range focuses on the first and third quartiles, which will be at the quarter and three-quarter points in the ordered series, and which can be calculated, respectively, by the following (2.8) and (2.9) The interquartile range is then given by the difference between the two (2.10) This measure of spread is usually considered superior to the range since it is not so heavily influenced by one or two extreme outliers that by definition would be right at the end of an ordered series and so would affect the range. However, the semi-interquartile range still only incorporates two of the observations in the entire sample. Variance and Standard Deviation Another, more familiar, measure of the spread or dispersion of a set of data, the variance, is very widely used. It is interpreted as the average squared deviation of each data point about the mean value, and is calculated using the usual formula for the variance of a sample for a variable y

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(2.11) A further measure of spread, the standard deviation, is calculated by taking the square root of the variance formula given in equation (2.11) (2.12) The squares of the deviations from the mean are taken rather than the deviations themselves to ensure that positive and negative deviations (for points above and below the average, respectively) do not cancel each other out. While there is little to choose between the variance and the standard deviation in terms of which is the best measure, the latter is sometimes preferred since it will have the same units as the variable whose spread is being measured, whereas the variance will have units of the square of the variable. So, for example, if yi are observations on the prices of houses in a particular region in thousands of UK pounds, then σ2 will have units of the square of prices (i.e., millions of pounds in this case) while σ will have units of thousands of pounds and so is more intuitive to interpret. Both variance and standard deviation share the advantage that they encapsulate information from all the available data points, unlike the range and quartile deviation, although they can also be heavily influenced by outliers (but to a lesser degree than the range). The quartile deviation is an appropriate measure of spread if the median is used to define the average value of the series, while the variance or standard deviation will be appropriate if the arithmetic mean constitutes the measure of central tendency adopted. Before moving on, it is worth discussing why the denominator in the formulae for the variance and standard deviation include N − 1 rather than N, the sample size. Subtracting one from the number of available data points is known as a degrees of freedom correction, and this is necessary since the spread is being calculated about the mean of the series, and this mean has had to be estimated from the sample data as well. Thus the spread measures described above are known as the sample variance and the sample standard deviation. Had we been observing the entire population of data rather than a mere sample from it, then the formulae would not need a degrees of freedom correction and we would divide by N rather than N − 1. 99

A further measure of dispersion is the negative semi-variance, which also gives rise to the negative semi-standard deviation. These measures use identical formulae to those described above for the variance and standard deviation, but when calculating their values, only those observations for which are used in the sum, and N now denotes the number of such observations. This measure is sometimes useful if the observations are not symmetric about their mean value (i.e., if the distribution is skewed – see the next section),4 and since they ignore deviations above the mean, they are sometimes used as measures of downside risk. The Coefficient of Variation A final statistic that has some uses for measuring dispersion is the coefficient of variation, CV. This is obtained by dividing the standard deviation by the arithmetic mean of the series (often multiplied by 100 to express it in percentage terms) (2.13) CV is useful where we want to make comparisons across series. Since the standard deviation has units of the series under investigation, it will scale with that series. Thus, if we wanted to compare the spread of monthly apartment rental values in London with those in Manchester, say, using the standard deviation would be misleading as the average rental value in London will be much bigger. By normalising the standard deviation, the coefficient of variation is a unit-free (dimensionless) measure of spread and so could be used more appropriately to compare series that have different scales.

2.3.3 Higher Moments If the observations for a given set of data follow a normal distribution, then the mean and variance are sufficient to entirely describe the series. In other words, it is impossible to have two different normal distributions with the same mean and variance. However, most samples of data do not follow a normal distribution, and therefore we also need what are known as the higher moments of a series to fully characterise them. The mean and the variance are the first and second moments of a distribution, respectively, and the (standardised) third and fourth moments are known as the 100

skewness and kurtosis, respectively. Skewness defines the shape of the distribution, and measures the extent to which it is not symmetric about its mean value. When the distribution of data is symmetric and unimodal (i.e., it only has one peak rather than many), the three methods for calculating the average (mean, mode and median) of the sample will be equal. If the distribution is positively skewed (where there is a long right hand tail and most of the data are bunched over to the left), the ordering will be mean > median > mode, whereas if the distribution is negatively skewed (a long left hand tail and most of the data bunched on the right), the ordering will be the opposite. A normally distributed series has zero skewness (i.e., it is symmetric). Kurtosis measures the fatness of the tails of the distribution and how peaked at the mean the series is. A normal distribution is defined to have a coefficient of kurtosis equal to three. It is possible to define a coefficient of excess kurtosis, equal to the coefficient of kurtosis minus three; a normal distribution will thus have a coefficient of excess kurtosis of zero. A normal distribution is said to be mesokurtic. Denoting the observations on a series by yi and their variance by σ2, it can be shown that the coefficients of skewness and kurtosis can be calculated respectively as5 (2.14) and (2.15) It is worth noting that, given the way that they are constructed, the skewness can be positive or negative while the kurtosis can only be positive (or zero), in the same way that a variance cannot be negative. To give some illustrations of what a series having specific departures from normality may look like, consider Figures 2.4 and 2.5. A normal distribution is symmetric about its mean, while a skewed distribution will not be, but will have one tail longer than the other (Figure 2.4).

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Figure 2.4 A normal versus a skewed distribution

Figure 2.5 A normal versus a leptokurtic distribution

A leptokurtic distribution is one which has fatter tails and is more peaked at the mean than a normally distributed random variable with the same mean and variance, while a platykurtic distribution will be less peaked in the mean, will have thinner tails, and more of the distribution in the shoulders than a normal. In practice, a leptokurtic distribution is more likely to characterise real estate (and economic) time series, and to characterise the residuals from a time-series model. In Figure 2.5, the leptokurtic distribution is shown by the blue line, with the normal by the dotted line. There is a formal test for normality, and this will be described and discussed in Chapter 5.

2.3.4 Measures of Association The summary measures we have examined so far have looked at each series in isolation. However, it is also very often of interest to consider the 102

links between variables. There are two key descriptive statistics that are used for measuring the relationships between series: the covariance and the correlation. Covariance The covariance is a measure of linear association between two variables and represents the simplest and most common way to enumerate the relationship between them. It measures whether they on average move in the same direction (positive covariance), in opposite directions (negative covariance), or have no association (zero covariance). The formula for calculating the covariance, σx,y, between two series, x and y, is given by (2.16) Correlation A fundamental weakness of the covariance as a measure of association is that it scales with the standard deviations of the two series, so it has units of x × y. Thus, for example, multiplying all of the values of series y by ten will increase the covariance tenfold, but it will not really increase the true association between the series since they will be no more strongly related than they were before the rescaling. The implication is that the particular numerical value that the covariance takes has no useful interpretation on its own and hence is not particularly useful. Therefore, the correlation takes the covariance and standardises or normalises it so that it is unit free. The result of this standardisation is that the correlation is bounded to lie on the (−1,1) interval. A correlation of 1 (−1) indicates a perfect positive (negative) association between the series. The correlation measure, usually known as the correlation coefficient, is often denoted ρx,y, and is calculated as (2.17) where σx and σy are the standard deviations of x and y, respectively. This measure is more strictly known as Pearson’s product moment correlation. To calculate Pearson’s correlation validly requires that the series are linearly related to one another and any formal hypothesis tests 103

involving this correlation measure would require the two series under study to be normally distributed. In cases where this does not apply, we can instead use Spearman’s rank correlation. As the name suggests, using this measure involves calculating the ranks of each element of the two separate series and then computing the correlation between the two series of ranks in the usual way. Spearman’s rank correlation is an example of a nonparametric test since it does not require any distributional assumptions (e.g., normality) to be validly applied. Copulas Covariance and correlation provide simple measures of association between series. However, as is well known, they are very limited measures in the sense that they are linear and are not sufficiently flexible to provide full descriptions of the relationship between financial series in reality. In particular, new types of assets and structures in finance have led to increasingly complex dependencies that cannot be satisfactorily modelled in this simple framework. Copulas provide an alternative way to link together the individual (marginal) distributions of series to model their joint distribution. One attractive feature of copulas is that they can be applied to link together any marginal distributions that are proposed for the individual series. The most commonly used copulas are the Gaussian and Clayton copulas. They are particularly useful for modelling the relationships between the tails of series, and find applications in stress testing and simulation analysis. For introductions to this area and applications in finance and risk management, see Nelsen (2006) and Embrechts et al. (2013).

2.3.5 An Example of How to Calculate Summary Statistics We now build an example that pulls together all of the material above on computing summary statistics. Suppose that we have annual data on the performance (annual returns in per cent) of two fund managers working for the same company, Risky Ricky and Safe Steve – you can see where this is going – for 13 years between 2005 and 2017. There were some scandals involving the company and investors withdrew their funds in large numbers so unfortunately one of the fund managers has to be made redundant to save money. Which of the two fund managers has shown the best performance and so should be retained? If we look at the data in Table 2.1, which shows the annual investment 104

returns on each of the two managers’ portfolios (in the first two columns after the years), it is clear that Ricky’s returns are more volatile – that is, they move up and down more – but it is not clear just by looking which fund manager has performed better taking into account all of the years. Ignore the final two columns of ranks for now. Table 2.1 Annual performance of two funds

Note: The first two data columns give the annual performance of each fund in percentage points for the 13-year sample period, while the final two columns provide the ranks of each year’s returns within the 13 annual return figures for each fund manager.

Figure 2.6 includes a time-series plot of the performance of the two managers by year in panel A (left-hand side), while panel B (right-hand side) presents a scatter plot of the returns of the two managers with Ricky on the x-axis scale and Steve on the y-axis so that we can see if the two lie roughly on a straight line. Having a look at the data and getting a feel for it before conducting more sophisticated analysis is called exploratory data analysis. It is very important to always plot and summarise the data before building any models as the preliminary analysis can often inform the research agenda for more sophisticated models and can avoid the kinds of mistakes that can arise when researchers go straight into autopilot’.

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Figure 2.6 A time-series plot and scatter plot of the performance of two

fund managers We can compute the summary statistics using the formulae above, either manually by plugging the numbers in and using a pocket calculator or by using a spreadsheet or econometrics package. If we use Excel and the data on Ricky’s annual performance is in cells C4 to C16, we can use the following four built-in Excel functions to compute the mean, standard deviation, skewness and kurtosis of the annual returns in cells C17 to C20 respectively as

Next, if the annual returns for Steve are in cells 4 to 16 of column D, we can drag the four resulting figures for the mean, standard deviation, skewness and kurtosis above across from column C into the next column D to calculate them for Steve. Note that the formula above will calculate the arithmetic average return rather than the geometric mean – whether this is the correct approach will depend on how the original annual return figures were calculated, as discussed earlier in this chapter. Excel has a function =GEOMEAN that can be used to calculate the geometric mean of a series, but this will only work when all of the numbers to be averaged are positive as it uses a different formula to equation (2.5) above. If we look at the mean returns, we can see that Ricky’s is a full percentage point higher (5.69 versus 4.69, with all figures expressed to two decimal places). But this higher mean for Ricky comes at the expense of Ricky having a much higher standard deviation of returns and it is clear from the plot in Figure 2.6 that his performance is much more volatile: doing better in good years and worse in bad years. 106

What about the higher moments? It is possible to show (see Scott and Horvath, 1980) that investors care about all moments of the return distribution, not just the first two. We know that investors prefer higher values of the first moment (the mean) and lower values of the second moment (the standard deviation)6 but Scott and Horvath demonstrate that investors also prefer higher values of all odd moments and lower values of all even moments, so that they want a higher skewness and a lower kurtosis. If we look at the higher moment values for Ricky and Steve, the results on who is the better performer are again inconclusive. The skewness figures are, respectively, 1.12 and 0.29, and although they are both positive, this favours Ricky. On the other hand, the kurtosis values are 2.10 and 1.46, and therefore Steve’s is better because it is lower and thus the distribution is more tightly centred around the mean (for given mean and variance). So the company owner has a dilemma if he or she only wants to retain one of the staff: Ricky has a better (higher) mean and skewness while Steve has a better (lower) standard deviation and kurtosis. The way to pick one of the two fund managers would be to construct a composite performance measure that includes information from more than one moment of the distribution simultaneously. The Sharpe ratio, which is calculated as the mean fund return (minus the risk-free rate of return) divided by the standard deviation of returns, is one such performance measure that is very popular. The Sharpe ratio is simple to calculate and very widely used but it only includes information from the first two moments of the portfolio return distribution. So strictly it is valid only if either investors care just about the first two moments (i.e., they ignore skewness and kurtosis) or if returns follow a normal distribution so that the skewness is zero and the kurtosis is the same as that of a normal distribution (equal to 3) – see above. In practice, however, neither of these two restricted cases are likely to apply and so it would be preferable to adopt a composite performance measure that incorporates information from the whole distribution rather than just the two moments, such as an appropriate utility function – see, for example, Brooks, Cerny and Miffre (2012) for a discussion. We might also be interested to know the relationship between the returns on Ricky’s fund and those of Steve’s fund. Do the values of the two portfolios tend to move together over time, which would be the case if they were investing in similar asset classes, albeit Ricky’s being more 107

volatile? We can get an idea of this by computing the correlation between the two sets of returns. In Excel, we can obtain covariances and correlations by using the =COVAR(C4:C16,D4:D16) and =CORREL(C4:C16,D4:D16) functions, respectively. As discussed above, covariances scale with the data and so are hard to interpret, but correlations must lie between −1 and +1. The Pearson correlation between Ricky’s and Steve’s sets of returns is 0.87, which is very close to one, indicating that the two series do move very closely together over time. This is also evident from the scatter plot on the right in Figure 2.6, where each return pairing lies close to a positive upward straight line. For interest, we also calculate the Spearman rank correlation measure. This is achieved by calculating the ranks for each member of the two return series – for example, to calculate the rank of the first observation in Ricky’s series, we would use the function =RANK(C4,C$4:C$16). We then drag this command down the column and drag that column across to create a similar set of ranks for Steve’s returns. The final two columns in Table 2.1 present the ranks for Ricky and Steve, respectively. As can be seen, again these are highly related. Finally, to calculate the rank correlation measure, we simply use the CORREL formula on the two columns of ranks, and this results in a calculated figure of 0.76, which is a bit lower than for the Pearson correlation, but nonetheless suggestive that the two series very much move together.

2.3.6 Useful Algebra for Means, Variances and Covariances There are several fairly straightforward equations that are useful for working with the expectations operator, the variance operator, and the covariance operator. In other words, these equations show how expressions for means, variances and covariances of random variables can be manipulated. The mean of a random variable y is also known as its expected value, written E(y). The properties of expected values are used widely in econometrics, and are listed below, referring to a random variable y The expected value of a constant (or a variable that is non-stochastic) is the constant, e.g., E(c) = c. The expected value of a constant multiplied by a random variable is equal to the constant multiplied by the expected value of the variable: E(c y) = c E(y). It can also be stated that E(c y + d) = c E(y) + d, 108

where d is also a constant. For two independent random variables, y1 and y2, E(y1y2) = E(y1) E(y2). The variance of a random variable y is usually written var(y). The properties of the ‘variance operator’, var(·), are The variance of a random variable y is given by var(y) = E[y − E(y)]2 The variance of a constant is zero: var(c) = 0 For c and d constants, var(c y + d) = c2 var(y) For two independent random variables, y1 and y2, var(c y1 + dy2) = c2var(y1) + d2var(y2). The covariance between two random variables, y1 and y2 may be expressed as cov(y1, y2). The properties of the covariance operator are cov(y1, y2) = E[(y1 − E(y1))(y2 − E(y2))] For two independent random variables, y1 and y2, cov(y1, y2) = 0 For four constants, c, d, e, and f, cov(c + dy1, e + fy2) = df cov(y1, y2).

2.4 Types of Data and Data Aggregation There are broadly three types of data that can be the employed in quantitative analysis of financial problems: time-series data, crosssectional data and panel data. Each of these will be discussed in turn next, but first it is worth mentioning another feature of data to watch out for, its degree of aggregation. Many data forms begin as individual observations but for various reasons are then aggregated. For example, we could measure the selling price of a specific house in a specific street every time it is sold and observe how and why it changes over time. But usually, a particular house is unlikely to be sold more often than once every five or ten years. So it is common to form house price indices, which measure the ‘average’ value of houses sold during a specific time period such as a month. Thus, the individual sales prices of a number of houses would be combined (aggregated) in some way and transformed into an index. House prices could be aggregated to the street level, the town level, the county level or the country level. All of these indices would get around the problem that specific houses are not sold very often by combining the information from sales of many different houses. However, the researcher 109

who constructs the index needs to be careful to compare like with like or to adjust the data in some way to account for the variations in the types of houses sold and to produce what are sometimes termed constant quality house price indices. We might be interested in looking at the national pattern – and in such cases we would want a national index – for example, we might want to know what factors are causing house prices to decline in the UK as a whole over a particular time period and thus the prices of individual properties or of prices in specific towns are irrelevant. Note that by using aggregate data, we can see the ‘big picture’, but we lose a lot of detail. To illustrate, it might be that prices are rising when averaged across the whole of the UK, but if we look at disaggregate data for England, Scotland, Wales and Northern Ireland, we might find that actually, only prices in England are rising while those in the other component countries are all falling. But since there are more houses sold in England than elsewhere, the overall average is rising.

2.4.1 Time-Series Data Time-series data, as the name suggests, are data that have been collected over a period of time on one or more variables. Time-series data have associated with them a particular frequency of observation or frequency of collection of data points. The frequency is simply a measure of the interval over, or the regularity with which, the data are collected or recorded. Box 2.2 shows some examples of time-series data. BOX 2.2 Time-series data Series Industrial production Government budget deficit Money supply The value of a stock

Frequency Monthly or quarterly Annually Weekly As transactions occur

A word on ‘As transactions occur’ in Box 2.2 is necessary. Many financial data do not start their life as being regularly spaced. For example, the price of common stock for a given company might be recorded to have changed whenever there is a new trade or quotation placed by the financial information recorder. Such recordings are very 110

unlikely to be evenly distributed over time – for example, there may be no activity between, say, 5 p.m. when the market closes and 8.30 a.m. the next day when it reopens; there is also typically less activity around the opening and closing of the market, and around lunch time. Although there are a number of ways to deal with this issue, a common and simple approach is to select an appropriate frequency, and use as the observation for that time period the last prevailing price during the interval. It is also generally a requirement that all data used in a model be of the same frequency of observation. So, for example, regressions that seek to estimate an arbitrage pricing model using monthly observations on macroeconomic factors must also use monthly observations on stock returns, even if daily or weekly observations on the latter are available. The data may be quantitative (e.g., exchange rates, prices, number of shares outstanding), or qualitative (e.g., the day of the week, a survey of the financial products purchased by private individuals over a period of time, a credit rating, etc.). Problems that could be tackled using time-series data How the value of a country’s stock index has varied with that country’s macroeconomic fundamentals How the value of a company’s stock price has varied when it announced the value of its dividend payment The effect on a country’s exchange rate of an increase in its trade deficit In all of the above cases, it is clearly the time dimension which is the most important, and the analysis will be conducted using the values of the variables over time.

2.4.2 Cross-Sectional Data Cross-sectional data are data on one or more variables collected at a single point in time. For example, the data might be on: A poll of usage of internet stockbroking services A cross-section of stock returns on the New York Stock Exchange (NYSE) A sample of bond credit ratings for UK banks Problems that could be tackled using cross-sectional data 111

The relationship between company size and the return to investing in its shares The relationship between a country’s GDP level and the probability that the government will default on its sovereign debt

2.4.3 Panel Data Panel (also sometimes known as longitudinal) data have the dimensions of both time series and cross-sections, e.g., the daily prices of a number of blue chip stocks over two years. The estimation of panel regressions is an interesting and developing area, and will be examined in detail in Chapter 11. Fortunately, virtually all of the standard techniques and analysis in econometrics are equally valid for time-series and cross-sectional data. For time-series data, it is usual to denote the individual observation numbers using the index t, and the total number of observations available for analysis by T. For cross-sectional data, the individual observation numbers are indicated using the index i, and the total number of observations available for analysis by N. Note that there is, in contrast to the time series case, no natural ordering of the observations in a cross-sectional sample. For example, the observations i might be on the price of bonds of different firms at a particular point in time, ordered alphabetically by company name. So, in the case of cross-sectional data, there is unlikely to be any useful information contained in the fact that Barclays follows Banco Santander in a sample of bank credit ratings, since it is purely by chance that their names both begin with the letter ‘B’. On the other hand, in a time-series context, the ordering of the data is relevant since the data are usually ordered chronologically. In this book, the total number of observations in the sample will be given by T even in the context of regression equations that could apply either to cross-sectional or to time-series data. A final type of data, which we could argue is slightly different to any of the above, is pooled cross-section and time-series data. This occurs when the variable of interest has both time-series and cross-sectional dimensions, but for some reason we do not use these features and instead simply combine all of the observations together. For example, we might have monthly data for ten years on the amount of profit that six different traders in a team made, but if we ignore the time ordering of the data and we also ignore which traders generated which profits and simply put all of the profit figures into a single unordered column this would be a pooled 112

sample. This would not be a panel of data since we would not be able to observe the performance of each individual trader in different months. Effectively, pooled data are treated as if they were simply a larger crosssectional sample.

2.4.4 Continuous and Discrete Data As well as classifying data as being of the time-series or cross-sectional type, we could also distinguish them as being either continuous or discrete, exactly as their labels would suggest. Continuous data can take on any value and are not confined to take specific numbers; their values are limited only by precision. For example, the rental yield on a property could be 6.2%, 6.24% or 6.238%, and so on. On the other hand, discrete data can only take on certain values, which are usually integers (whole numbers), and are often defined to be count numbers.7 For instance, the number of people in a particular underground carriage or the number of shares traded during a day. In these cases, having 86.3 passengers in the carriage or 58571/2 shares traded would not make sense. The simplest example of a discrete variable is a Bernoulli or binary random variable, which can only take the values 0 or 1 – for example, if we repeatedly tossed a coin, we could denote a head by 0 and a tail by 1.

2.4.5 Cardinal, Ordinal and Nominal Numbers Another way in which we could classify numbers is according to whether they are cardinal, ordinal or nominal. Cardinal numbers are those where the actual numerical values that a particular variable takes have meaning, and where there is an equal distance between the numerical values. On the other hand, ordinal numbers can only be interpreted as providing a position or an ordering. Thus, for cardinal numbers, a figure of 12 implies a measure that is ‘twice as good’ as a figure of 6. Examples of cardinal numbers would be the price of a share or of a building, and the number of houses in a street. On the other hand, for an ordinal scale, a figure of 12 may be viewed as ‘better’ than a figure of 6, but could not be considered twice as good. Examples of ordinal numbers would be the position of a runner in a race (e.g., second place is better than fourth place, but it would make little sense to say it is ‘twice as good’) or the level reached in a computer game. The final type of data that could be encountered would be where there is no natural ordering of the values at all, so a figure of 12 is simply different 113

to that of a figure of 6, but could not be considered to be better or worse in any sense. Such data often arise when numerical values are arbitrarily assigned, such as telephone numbers or when codings are assigned to qualitative data (e.g., when describing the exchange that a US stock is traded on, ‘1’ might be used to denote the NYSE, ‘2’ to denote the NASDAQ and ‘3’ to denote the AMEX). Sometimes, such variables are called nominal variables. Cardinal, ordinal and nominal variables may require different modelling approaches or at least different treatments, as should become evident in the subsequent chapters.

2.5 Arithmetic and Geometric Series A series or a sequence is simply a list of numbers in a particular order. An arithmetic series, also known as an arithmetic progression, is a sequence where a specific entry in that series is formed by adding a fixed number, known as the common difference, to the previous one. For example

The first of these is an arithmetic series with an initial value of 2 and adds 3 each time we move from one entry in the sequence to the next; the second row is an arithmetic series with an initial value of −10 and a common difference of −20. Arithmetic series do not have many uses in finance so we will not consider them further. A geometric series (geometric progression), on the other hand, is a series where instead of adding a fixed amount to move from one entry to the next, we multiply by a fixed amount (the common ratio). For example

The first of these is a geometric series with an initial value of 4 and a common ratio of 2, while the second row is a geometric series with an initial value of 2 and a common ratio of 0.5. Geometric series are very useful in finance as they describe the situation where a sum of money is invested and earns a certain percentage of interest in each time period. To develop some notation, let a denote the initial value of a geometric series (starting with the term numbered 0 and ending with term numbered n − 1), and let d denote the common ratio. Then we could write a geometric series containing n terms as 114

There is an expression that can be used to calculate the sum of the first n terms, denoted Sn, of the series (running from a to adn−1) (2.18) For instance, if a geometric series begins with 2 and has a common ratio of 3, the sum of the first 8 terms would be (2.19) As an exercise, calculate each of the first eight terms in this series and confirm that the sum is indeed 6560. Of particular use in financial applications is the infinite sum of a geometric progression, denoted S∞. We can see what will happen to as n tends to infinity, since the term dn will tend towards zero (so long as 0 < d < 1), in which case the expression can be simplified as (2.20) Here, even though there is an infinite number of terms in the series, their sum is finite. Note that if d ≥ 1, the series would not ‘converge’ (i.e., successive terms would not become smaller and smaller) and therefore the sum would be infinite.

2.6 Future Values and Present Values A fundamental concept in economics and finance is the notion that money has time value. This means that receipt of a given amount of money is worth a different amount depending on when it is received. In general, money has positive time value. This means that £100, for instance, is worth more if received today than next week, and worth more if received next week than if received next year. This arises for several reasons: cashflows are usually considered more risky the further in the future they are to be received (more time for something to go wrong!), inflation will erode the value of a fixed amount received in the future, and people have positive time preference, which means simply that they are impatient and would rather have stuff now than waiting until the future. 115

As a result of the time value of money, we cannot simply combine cashflows in their raw form into financial calculations if they are received at different points in time. The way that we ensure we are comparing likewith-like is to transform the cashflows to what they would be worth if they were all received at the same point in time. So we either transform all cashflows to the amount that they would be worth at some given point in the future (the future value) or we transform all future cashflows into the equivalent amount that they would be worth if received today (the present value). If we transform current values into future ones we are compounding, while if we transform future values into present ones we are discounting. We will now look at each of these two concepts in turn and examine how they are used.

2.6.1 Future Values Suppose that we place £100 in a bank savings account for five years, paying an annual interest rate of 2%. The sum of money in the account at the end of the period would be given by (2.21) where PT denotes the terminal (future) value of the account, r is the interest rate (expressed as a proportion – e.g., 0.02, rather than a percentage), P0 is the amount placed in the account now, and T is the number of time periods for which the money is invested. In this example, the future value of the investment at the end of the first year would be PT = £100 × (1 + 0.02) = £102, while at the end of the second year it would have grown to PT = £100 × (1 + 0.02)2 = £102 × (1 + 0.02) = £104.04. The savings balance would continue to grow in this way until, at the end of the fifth year, it would have reached PT = £100 × (1 + 0.02)5 = £110.41. We would say in this case that the interest is compounded annually – in other words, interest is paid this year on the total value of this year’s end savings, which will comprise both last year’s savings value and last year’s interest. So after the first year, the saver earns interest on their previous interest as well as on the amount invested. This is why the saver earned £2 in interest in year one but £2.04 in year two: the extra 4p in year two is the additional interest on the £2 that had been earned in year one. We can rearrange the future value formula in equation (2.21) above to 116

make r the subject and this would enable us to calculate the rate of interest required to secure a specific future value, PT, given the initial investment P0 (2.22) For example, if we make an initial investment of £1000 and no further investments, and we leave the funds for ten years, what rate of interest is required to enable us to achieve a sum of £1500 by the end of the decade? The calculation is (2.23) So an annual interest rate of about 4.14% is required. A further re-arrangement of equations (2.21) and (2.22) enables us to make the term of the investment, T, the subject of the formula (2.24) So, for instance, if we can invest £1000 initially and wish it to grow to £2000, assuming an interest rate of 10% (we should be so lucky!), how many years do we need to wait? We would have (2.25) Notice that we can use the formula (2.24) above to determine how many years it would take for an investment to grow by a factor of Z (2.26) where PT = ZP0. So, for example, if we wanted to triple the initial investment, assuming that the interest rate remained at 10%, we would set Z = 3 and we would have (2.27)

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Thus about eleven an a half years are required – clearly a long time unless we could achieve an even higher interest rate! All of the above examples assume that interest is paid annually at the end of the year. But now suppose instead that the account paid an annual rate of interest of 2% with the payments made every six months (i.e., 1% paid every six months). Many companies pay dividends and many bonds pay coupons semi-annually so this is empirically relevant. In this case, the compounding would be semi-annual rather than annual. We would be better off since we would receive interest in the second six months of each year on the interest paid in the first six months. This effect would be very small with such a low interest rate, but we would calculate the future value for the first example above in this sub-section by now using an interval of six months, an interest rate r of 1%, and a number of periods T of 10 (i.e., ten periods each of six months): PT = £100 × (1 + 0.01)10 = £110.46 – not much extra interest to get excited about. If the interest was paid (compounded) monthly, the terminal value would be PT = £100 × (1 + (0.02/12))60 = £110.51. We can see that the higher the compounding frequency, the more interest would actually be received for a given nominal interest rate of 2%. We would call all of these situations where the compounding takes place discretely a simple interest calculation. In the above example, the nominal interest rate is 2% per annum but if the interest is compounded more frequently than annually, the actual interest rate received, known as the effective interest rate, will be higher. We can calculate this interest rate on an annualised basis simply as (2.28) where r is the nominal rate and n is the number of compounding periods per year. EXAMPLE 2.2 What is the effective rate when the nominal rate, r, is 2% and interest is compounded monthly (n = 12)? It would be: effective (to two decimal places).

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Another useful formula is one that calculates the future value of an investment (PT) when interest of r per year in total is paid n times per year for a T years on the original amount P0 (2.29) In the limit, as the compounding frequency increases and so we have more and more shorter and shorter time periods (i.e., we move from annual to monthly to weekly to daily to hourly compounding and so on), we would eventually reach a situation where the time period was infinitesimally small. We would term this continuous compounding. If interest is compounded continuously at an annual equivalent rate r, we would write (2.30) where e is the exponential number discussed in section 1.5.5 of Chapter 1. If T = 5 and r = 2%, then the terminal value if interest is continuously compounded is PT = e0.02×5 = £110.52. This is barely any different to the terminal value when interest is earned monthly but the difference would be more noticeable if r was higher, as the examples in Table 2.2 show. Table 2.2 Impact of different compounding frequencies on the effective interest rate and terminal value of an investment

Analogous to the formulae (2.22) and (2.24) above for simple interest calculations, we can re-arrange the expression (2.30) to make the continuously compounded interest rate the subject of the formula

119

(2.31) And we can make the number of years of investment the subject of the formula (2.32) Table 2.2 shows the effects of different interest rates and compounding frequencies on the terminal value of a £100 investment. Two results can clearly be seen from the table. The first is that the effect of compounding is stronger the higher the nominal interest rate since the additional benefit from getting some of the interest payment early and reinvesting it is greater in such cases. The second observable feature is that the incremental effect of further increasing the compounding frequency gradually reduces. For example, going from annual to quarterly compounding has a bigger effect even than going from quarterly to continuous compounding.

2.6.2 Present Value The reverse of calculating the future value of an amount of money that is earning interest in a bank account would be where we calculate the present value of an amount of money to be received at some point in the future. Instead of an interest rate as we would have for a future value calculation, in the case of present values we use a discount rate, which is the rate at which we would reduce the future payment into today’s terms. We would write (2.33) where P0 is the present value, r is now the discount rate, PT is the sum to be paid or received in the future, and T is the number of periods into the future that it will be paid or received. EXAMPLE 2.3 What is the present value of £100 to be received in five years’ time if the discount rate is 2%? This would be P0 = £100/(1 + 0.02)5 = £90.57. This shows that £100 120

in five years’ time is worth £90.57 in today’s money terms. Such present value calculations underpin most of the valuation models employed in finance as the situation is that investors purchase assets now and receive cashflows in the future, which then need to be converted into today’s terms (i.e., discounted back to the present) so that the amount to be paid for the asset now and the amount received in the future can be compared in equivalent terms. EXAMPLE 2.4 To illustrate a finance application, suppose we have a bond that pays a £5 coupon annually with the next coupon due immediately, the bond has exactly five years left to maturity when it will be redeemed at its par value of £100 and an appropriate discount rate is 10%. What would be a fair price to pay today for the bond? We would calculate the fair price as the discounted sum of the six coupon payments (one now and one at the end of each of the next five years) and plus the discounted value of the par amount of the bond. If we let the fair price in pounds be denoted by P0, the calculation would be

(2.34)

We can think of the discounted values of the coupons as a geometric progression where each term is multiplied by (1/(1 + 0.1)) to get the next term. The sum of the coupons (noting that there are n = 6 of them rather than 5 since one is due immediately and d = 1/(1 + 0.1)) would be (2.35) We then need to calculate the present value of the redemption amount, which is 100/(1.1)5 = £62.09. Thus the fair price to pay for the bond is P0 = £23.95+£62.09 = £86.04. In fact, the coupon payments made on the bond are an example of an annuity, which is a financial product paying a fixed amount every period 121

for a fixed length of time. Many people choose (or are required by law) to purchase a specific type of annuity with their pension savings. This sort of annuity involves an insurance element, which guarantees to continue to pay the fixed amount for as long as the person lives (and thus provides insurance against the possibility that the individual will live a long time and run out of money), and the payments made might also increase with inflation during the time that the annuity is paying out. The formula to calculate the present value of an annuity paying a fixed amount a every period for T periods assuming a discount rate of r is (2.36) To demonstrate that this formula works, we can calculate the present value of the annuity represented by the coupon payments in the bond example above and show that it is indeed £23.95. (2.37) Note that the formula implicitly assumes that the first payment is made at the end of the first period – hence there are five of them, and then we need to add the first immediate £5 payment, which is not discounted. If the bond under study was irredeemable (so that it had an infinite lifetime like a stock), we would need to use the S∞ formula (2.20) to calculate its present value. As an illustration, what should an investor be willing to pay today for a perpetual (irredeemable) bond that pays a coupon of £5 every six months if the appropriate rate to discount future cashflows is 4% and the next coupon will be paid immediately? In this case, we could discount each cashflow with the discount rate for six months, which would be 2%. We could write this as an infinite series beginning with £5:

We need to be slightly careful as the common difference here is 1/(1 + 0.02) Then we would have the value as (2.38)

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Note that if the first coupon were not to be paid until the end of the first period, i.e., in six months’ time rather than immediately, the series would be:

and we could calculate the value as (2.39) So the £5 to be received immediately has now been removed, which reduces the present value by exactly £5 as this cashflow is not discounted. In the UK, irredeemable government bonds are sometimes known as consols. Sometimes, the payment amount may be growing over time rather than being fixed. For example, if we purchase shares in a company, the dividend that it pays will usually rise over time. If we assume that the rate of increase in the value of the dividend is some constant proportion g, this makes valuing the company much easier. So if we buy one share in the company now, we would receive a dividend in every period (assume that this is one year and that the first payment of D is due in exactly one year) in perpetuity (i.e., for ever), so the present value formula would be (2.40) Examining this formula, we can see that the value of the dividend is growing at a rate g but being reduced (discounted) at a rate r. If g > r, the present value of the future dividends would be growing over time and the share would have infinite value. Thus, for this sum to be convergent and for the share to have a finite value, we require g < r. If that is the case, we can calculate the infinite sum from equation (2.20) as (2.41) This is often known as the Gordon growth model of equity valuation based on the expected future stream of dividend payments. Finally, to complete this section, we should also note that analogous to continuous compounding, cashflows can be continuously discounted. The 123

formula would be (2.42) So, for example, we could calculate the present value of £100 to be received in five years’ time with an annual discount rate of 2% and continuous discounting as P0 = £100 × e−0.02×5 = £90.48

2.6.3 Internal Rate of Return It is sometimes the case that we know both the present value of a particular set of cashflows, and we know all of the future cashflows, but we do not know the discount rate, or in other words the rate of interest which implicitly the financial product would generate for us if we purchased it today. The value of r that would equate the amount to be paid today P0 to the present value of all of the cashflows we would receive if we purchased the asset is known as the internal rate of return or IRR. We could calculate that by solving the annuity formula above for r. More generally, if the future cashflow payments were not fixed but varied over time, we would have a more flexible formula (2.43) So the situation is that we know the value of P0, all values of ai (i = 1, …, T) and T but we want to find r. In general, there will be more than one solution to this equation – in other words, more than one value of r that sets the left- and righthand sides of this equation to be equal. If T = 1 or 2, then we would have a linear or a quadratic equation, respectively, to solve for r, which we could do analytically using the formulae we have learned in Section 1.5 of Chapter 1 on functions. But for T bigger than 2, the equation would be solved numerically. EXAMPLE 2.5: Calculating an Internal Rate of Return with Excel It is very straightforward to calculate internal rates of return using recent versions of Microsoft Excel. One way to do this would be to set out the cashflows in a spreadsheet, calculate their discounted values with a particular interest rate r and then use Solver to estimate the IRR. An alternative approach is to employ the IRR function that is built into Excel. For example, suppose that we purchase a bond today for £110 124

which has exactly five years to maturity when it will be redeemed at its par value of £100 and which will provide an annual coupon of £5. What is the internal rate of return of the bond investment (which is effectively the yield to maturity of the bond)? If we set up the spreadsheet so that the following entries are in cells A1 to B7: Year Cashflow 0 -107 1 5 2 5 3 5 4 5 5 105 Then in any other cell, we simply write the command where the second term in parentheses, 0.1, is an initial guess of the expected interest rate. The initial guess is required for situations where there are multiple IRRs in order that Excel chooses the most plausible value from among them. Then hitting ENTER leads Excel to calculate the IRR, which is 3.45% in this case. Multiple IRRs will occur for projects where the cashflows change sign more than once during the lifetime of the project. In the example just presented, the cashflow is only negative (an outflow) during year 0 and then positive (i.e., cash inflows only) always thereafter. But if we had further outflows during the project, the IRR would be non-unique. More specifically, we would have as many IRRs as there are cashflow sign changes. To illustrate, suppose that we now have the following cashflows for a project Year Cashflow 0 -100 1 240 2 -143 There are cash outflows in years 0 and 2 with an inflow in year 1. If we use the formula as above but reducing the cell range as we now only have three cashflows and still with a guess of 0.1 125

then we find the interest rate as 10.00%, but if instead we use an initial guess of 0.5 (so typing =IRR(B2:B4,0.5), then we would end up with an interest rate of 30.00%. What has happened here is that both interest rate values would solve the equation and would set the net present value (NPV) of the project to zero and so Excel will converge upon the answer that is closest to the initial value. More generally, it is possible for one or more of the calculated IRR values to be negative.

2.7 Returns in Financial Modelling In many of the problems of interest in finance, the starting point is a time series of prices – for example, the prices of shares in Ford, taken at 4 p.m. each day for 200 days. For a number of statistical reasons, it is preferable not to work directly with the price series, so that raw price series are usually converted into series of returns. Additionally, returns have the added benefit that they are unit-free. So, for example, if an annualised return were 10%, then investors know that they would have got back £110 for a £100 investment, or £1,100 for a £1,000 investment, and so on. There are two methods used to calculate returns from a series of prices, and these involve the formation of simple returns, and continuously compounded returns, which are, respectively,

(2.44)

(2.45)

where: Rt denotes the simple return at time t, rt denotes the continuously compounded return at time t, pt denotes the asset price at time t and ln denotes the natural logarithm. In the limit, as the frequency of the sampling of the data is increased so that they are measured over a smaller and smaller time interval, the simple and continuously compounded returns will be identical. If the asset under consideration is a stock or portfolio of stocks, the total return to holding it is the sum of the capital gain and any dividends paid 126

during the holding period. Usually, the holding period is one year but it could be any amount of time. However, researchers often ignore any dividend payments. This is unfortunate, and will lead to an underestimation of the total returns that accrue to investors. This is likely to be negligible for very short holding periods, but will have a severe impact on cumulative returns over investment horizons of several years. Ignoring dividends will also have a distortionary effect on the crosssection of stock returns. For example, ignoring dividends will imply that ‘growth’ stocks with large capital gains will be inappropriately favoured over income stocks (e.g., utilities and mature industries) that pay high dividends. Alternatively, it is possible to adjust a stock price time series so that the dividends are added back to generate a total return index. If pt were a total return index, returns generated using either of the two formulae presented above thus provide a measure of the total return that would accrue to a holder of the asset during time t. The academic finance literature generally employs the log-return formulation (also known as log-price relatives since they are the log of the ratio of this period’s price to the previous period’s price). Box 2.3 shows two key reasons for this. BOX 2.3 Log returns (1) Log returns have the nice property that they can be interpreted as continuously compounded returns – so that the frequency of compounding of the return does not matter and thus returns across assets can more easily be compared. (2) Continuously compounded returns are time-additive. For example, suppose that a weekly returns series is required and daily log returns have been calculated for five days, numbered 1 to 5, representing the returns on Monday through Friday. It is valid to simply add up the five daily returns to obtain the return for the whole week: Monday return

r1 = ln (p1/p0) = ln p1 − ln p0

Tuesday return

r2 = ln (p2/p1) = ln p2 − ln p1

Wednesday return

r3 = ln (p3/p2) = ln p3 − ln p2

Thursday return

r4 = ln (p4/p3) = ln p4 − ln p3

Friday return

r5 = ln (p5/p4) = ln p5 − ln p4 127

___________ ln p5 − ln p0 = ln (p5/p0)

Return over the week

There is, however, also a disadvantage of using the log returns. The simple return on a portfolio of assets is a weighted average of the simple returns on the individual assets (2.46) But this does not work for the continuously compounded returns, so that they are not additive across a portfolio. The fundamental reason why this is the case is that the log of a sum is not the same as the sum of a log, since the operation of taking a log constitutes a non-linear transformation. Calculating portfolio returns in this context must be conducted by first estimating the value of the portfolio at each time period and then determining the returns from the aggregate portfolio values. Or alternatively, if we assume that the asset is purchased at time t − K for price pt−K and then sold K periods later at price pt, then if we calculate simple returns for each period, Rt, Rt + 1, …, RK, the aggregate return over all K periods is

(2.47)

This is known as the holding period return, where the last line uses the ∏ notation presented in Section 1.5.9 from Chapter 1. The annualised holding period return (call it RH) is given by the Kth root of equation (2.47) minus one (2.48) EXAMPLE 2.6

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Given the data in the following table, calculate the returns for each year on a single share in a company that is purchased on 31 December 2012 for a price of 100 pence and then held for four years. Next, calculate the total return over the whole period and the annualised average return. Date 31 Dec 2012 31 Dec 2013 31 Dec 2014 31 Dec 2015 31 Dec 2016

Price 100p

Dividend -

120p

10p

130p

10p

140p

10p

167p

10p

SOLUTION Assume here that we use the simple return formula rather than continuously compounded returns. The first step is to calculate each year’s return separately for the calendar years 2013 (31 Dec 2012 – 31 Dec 2013), 2014, 2015 and 2016. Using the notation R13 to denote the 2013 return and so on, we have R13 = 100 × (120 − 100 + 10)/100 = 30%, R14 = 100 × (130 − 120 + 10)/120 = 16.7%, R15 = 100 × (140 − 130 + 10)/130 = 15.4%, and R16 = 100 × (167 − 140 + 10)/140 = 26.4%. Next, we calculate the holding period return for the whole four years as

So the return over the whole period, RK,t, is 2.213 − 1 = 1.213 or 121.3%. This figure can be annualised by taking the fourth root of one plus it and then subtracting one – in other words, by calculating the geometric mean of the series of individual returns

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RH, the average annual holding period return, is thus around 0.22 or 22%.

2.7.1 Real versus Nominal Series and Deflating Nominal Series If a newspaper headline suggests that ‘house prices are growing at their fastest rate for more than a decade. A typical 3-bedroom house is now selling for £280,000, whereas in 2005 the figure was £120,000’, it is important to appreciate that this figure is almost certainly in nominal terms. That is, the article is referring to the actual prices of houses that existed at those points in time. The general level of prices in most economies around the world has a general tendency to rise almost all of the time, so we need to ensure that we compare prices on a like-for-like basis. We could think of part of the rise in house prices being attributable to an increase in demand for housing, and part simply arising because the prices of all goods and services are rising together. It would be useful to be able to separate the two effects, and to be able to answer the question, ‘how much have house prices risen when we remove the effects of general inflation?’ or equivalently, ‘how much are houses worth now if we measure their values in 1990-terms?’ We can do this by deflating the nominal house price series to create a series of real house prices, which is then said to be in inflation-adjusted terms or at constant prices. Deflating a series is very easy indeed to achieve: all that is required (apart from the series to deflate) is a price deflator series, which is a series measuring general price levels in the economy. Series like the consumer price index (CPI), producer price index (PPI) or the GDP Implicit Price Deflator, are often used. A more detailed discussion of which is the most relevant general price index to use is beyond the scope of this book, but suffice to say that if the researcher is only interested in viewing a broad picture of the real prices rather than a highly accurate one, the choice of deflator will be of little importance. The real price series is obtained by taking the nominal series, dividing it by the price deflator index, and multiplying by 100 (under the assumption that the deflator has a base value of 100) (2.49) It is worth noting that deflation is only a relevant process for series that are measured in money terms, so it would make no sense to deflate a quantity-based series such as the number of shares traded or a series 130

expressed as a proportion or percentage, such as the rate of return on a stock. EXAMPLE 2.7: DEFLATING HOUSE PRICES Let us use for illustration a series of average UK house prices, measured annually for 2006 – 18 and given in column 2 of Table 2.3. Some figures for the general level of prices as measured by the CPI are given in the column 3. So first, suppose that we want to convert the figures into constant (real) prices. Given that 2009 is the ‘base’ year (i.e., it has a value of 100 for the CPI), the easiest way to do this is simply to divide each house price at time t by the corresponding CPI figure for time t and then multiply it by 100, as per equation (2.49). This will give the figures in column 4 of the table. Table 2.3 How to construct a series in real terms from a nominal one

Notes: All prices in British pounds; house price figures and CPI are for illustration only.

If we wish to convert house prices into a particular year’s figures, we would apply equation (2.49), but instead of 100 we would have the CPI value that year. Consider that we wished to express nominal house 131

prices in 2018 terms (which is of particular interest as this is the last observation in the table). We would thus base the calculation on a variant of (2.49) (2.50) So, for example, to get the 2006 figure (i.e., t is 2006) of 105,681 for the average house price in 2018 terms, we would take the nominal figure of 83,450, multiply it by the CPI figure for the year that we wish to make the price for (the reference year, 123.6) and then divide it by the CPI figure for the year 2006 (97.6). Thus etc.

2.8 Portfolio Theory Using Matrix Algebra Probably the most important application of matrix algebra in finance is solving portfolio allocation problems. Although these can be solved in a perfectly satisfactory fashion with sigma notation rather than matrix algebra, use of the latter does considerably simplify the expressions and makes it easier to solve them when the portfolio includes more than two assets. This book is not the place to learn about portfolio theory per se – interested readers are referred to Bodie, Kane and Marcus (2014) or the many other investment textbooks that exist – rather, the purpose of this section is to demonstrate how matrix algebra is used in practice, drawing together the material in Chapter 1 together with what we have learned in this chapter on computing means, variances and covariances in a practical application. So, let us pick up from the material we covered in Section 1.7 in Chapter 1 now that we have also covered the construction of means, variances, covariances and returns. To start, suppose that we have a set of N stocks that are included in a portfolio P with weights w1, w2, …, wN and suppose that their expected returns are written as E(r1), E(r2), …, E(rN). We could write the N × 1 vectors of weights, w, and of expected returns, E(r), as

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For instance, w3 and E(r3) are the weight attached to stock three and its expected return, respectively. The expected return on the portfolio, E(rP) can be calculated as E(r)′w – that is, we multiply the transpose of the expected return vector by the weights vector. We then need to set up what is called the variance–covariance matrix of the returns, denoted V. This matrix includes all of the variances of the components of the portfolio returns on the leading diagonal and the covariances between them as the off-diagonal elements. We will also discuss such a matrix extensively in Chapter 4 in the context of the parameters from regression models. The variance–covariance matrix of the returns may be written

The elements on the leading diagonal of V are the variances of each of the component stocks’ returns - so, for example, σ11 is the variance of the returns on stock one, σ22 is the variance of returns on stock two and so on. The off-diagonal elements are the corresponding covariances – so, for example, σ12 is the covariance between the returns on stock one and those on stock two, σ58 is the covariance between the returns on stock five and those on stock eight, and so on. Note that this matrix will be symmetrical about the leading diagonal since cov(a, b) = cov(b, a) where a and b are random variables, and hence it is possible to write σ12 = σ21 and so forth. In order to construct a variance–covariance matrix, we would need to first set up a matrix containing observations on the actual returns (not the expected returns) for each stock where the mean, has been subtracted away from each series i. If we call this matrix R, we would write

So each column in this matrix represents the deviations of the returns on individual stocks from their means and each row represents the meanadjusted return observations on all stocks at a particular point in time. The 133

general entry, rij, is the jth time-series observation on the ith stock. The variance–covariance matrix would then simply be calculated as V = (R ′R)/(T − 1) where T is the total number of time-series observations available for each series. Suppose that we wanted to calculate the variance of returns on the portfolio P (a scalar which we might call VP). We would do this by calculating (2.51) Checking the dimension of VP, w′ is (1 × N), V is (N × N) and w is (N × 1) so VP is (1 × N × N × N × N × 1), which is (1 × 1) as required. We could also define a correlation matrix of returns, C, which would be

This matrix would have ones everywhere on the leading diagonal (since the correlation of something with itself is always one) and the off-diagonal elements would give the correlations between each pair of returns – for example, C35 would be the correlation between the returns on stock three and those on stock five. Note again that, as for the variance–covariance matrix, the correlation matrix will always be symmetrical about the leading diagonal so that C31 = C13 etc. Using the correlation instead of the variance–covariance matrix, the portfolio variance given in equation (2.51) would be (2.52) where C is the correlation matrix, w is again the vector of portfolio weights, and S is a diagonal matrix with each element containing the standard deviations of the portfolio returns. Selecting Weights for the Minimum Variance Portfolio Although in theory investors can do better by selecting the optimal portfolio on the efficient frontier, in practice a variance minimising portfolio often performs well when used out-of-sample. Thus we might 134

want to select the portfolio weights w that minimise the portfolio variance, VP. In matrix notation, we would write

We also need to be slightly careful to impose at least the restriction that all of the wealth has to be invested otherwise this minimisation problem can be trivially solved by setting all of the weights to zero to yield a zero portfolio variance. This restriction that the weights must sum to one is written using matrix algebra as w′ · 1N = 1, where 1N is a column vector of ones of length N.8 The minimisation problem can be solved to (2.53) where MVP stands for minimum variance portfolio. Selecting Optimal Portfolio Weights In order to trace out the mean–variance efficient frontier, we would repeatedly solve this minimisation problem, but in each case set the portfolio’s expected return equal to a different target value, So, for example, we set to 0.1 and find the portfolio weights that minimise VP, then set to 0.2 and find the portfolio weights that minimise VP, and so on. We would write this as

This problem is sometimes called the Markowitz portfolio allocation problem, and can be solved analytically as expressed above. That is, we can derive an exact solution using matrix algebra. However, it is often the case that we want to place additional constraints on the optimisation – for instance we might want to restrict the portfolio weights so that none are greater than 10% of the overall wealth invested in the portfolio, or we might want to restrict them to all be positive (i.e., long positions only with no short selling allowed). In such cases, the Markowitz portfolio allocation problem cannot be solved analytically and thus a numerical procedure must be used such as the Solver function in Microsoft Excel. Note that it is also possible to write the Markowitz problem the other way around – that is, where we select the portfolio weights that maximise 135

the expected portfolio return subject to a target maximum variance level. If the procedure above is followed repeatedly for different return targets, it will trace out the efficient frontier. In order to find the tangency point where the efficient frontier touches the capital market line, we need to solve the following problem

If no additional constraints are required on the stock weights, this can be solved fairly simply as (2.54)

2.8.1 The Mean–Variance Efficient Frontier in Excel This section will now describe how to construct an efficient frontier and draw the capital market line using a three-stock portfolio with Microsoft Excel. It is assumed that the reader knows the standard functions of Excel – for those who need a refresher, see the excellent book by Benninga (2017). The spreadsheet ‘efficient.xls’ contains the finished product – the plots of the efficient frontier and capital market line. However, I suggest starting with a blank spreadsheet, copying across the raw data and starting to reconstruct the formulae again to get a better of idea of how it is done. The first step is to construct the returns. The raw prices and T-bill yields are in columns two to six of the sheet. We are going to assume a threeasset portfolio. However, all of the principles outlined below could be very easily and intuitively extended to situations where there were more assets employed. Since we are dealing with portfolios, it is probably preferable to employ simple rather than continuously compounded returns. So start by constructing three sets of returns for the Ford, General Electric and Microsoft share prices in columns H to J, and head these columns ‘FORDRET’, ‘GERET’ and ‘MSOFTRET’, respectively. Column K will comprise the weights on a portfolio containing all three stocks but with varying weights. The way we achieve this is to set up three cells that will contain the weights. To start with, we fix these arbitrarily but later will allow the Solver to choose them optimally. So write 0.33, 0.33 and 0.34 136

in cells N12 to N14, respectively. In cell N15, calculate the sum of the weights as a check that this is always one so that the all wealth is invested among the three stocks. We are now in a position to construct the (equally weighted) portfolio returns (call them ‘PORTRET’) in column K. In cell K2, write =H3*$N$12+I3*$N$13+J3*$N$14 and then copy this formula down the whole of column K until row 137. The next stage is to construct the variance–covariance matrix, which we termed V in the description above. So first, click on Data and Data Analysis and then select Covariance from the menu. Complete the Window so that it appears as in screenshot 2.1 with input range $H$3:$J$137 and output range $M$3:$P$6 and click OK.

Screenshot 2.1 Setting up a variance–covariance matrix in Excel

Now copy the covariances so that they are also in the upper right triangle of the matrix, and also replace ‘Column 1’ etc. with the names of the three stocks in the column and row headers. We now want to calculate the average returns for each of the individual stocks (we already have their variances on the leading diagonal of the variance−covariance matrix). To do this, in cells M9 to O9, write =AVERAGE(H3:H137), =AVERAGE(I3:I137) and =AVERAGE(I3:I137). Next, we can construct summary statistics for the portfolio returns. There are several ways to do this. One way would be to calculate the mean, variance and standard deviation of the returns directly from the monthly portfolio returns in column K. However, to see how we would do 137

this using matrix algebra in Excel, for calculating the average portfolio return in cell N18, enter the formula =MMULT(M9:O9,N12:N14) which will multiply the returns vector (what we called E(r)′) in M9 to O9 by the weights vector w in N12 to N14. In cell N19, we want the formula for the portfolio variance, which is given by w′Vw and in Excel this is calculated using the formula =MMULT(MMULT(Q13:S13, N4:P6),N12:N14). Effectively, we are conducting the multiplication in two stages. First, the internal MMUL is multiplying the transposed weights vector, w′ in Q13 to S13 by the variance−covariance matrix V in N4 to P6. We then multiply the resulting product by the weights vector w in N12 to N14. Finally, calculate the standard deviation of the portfolio returns in N19 as the square root of the variance in N18. Take a couple of minutes to examine the summary statistics and the variance−covariance matrix. It is clear that Ford is by far the most volatile stock with an annual variance of 239, while Microsoft is the least at 50. The equally weighted portfolio has a variance of 73.8. Ford also has the highest average return. We now have all of the components needed to construct the mean–variance efficient frontier and the right-hand side of your spreadsheet should appear as in Screenshot 2.2.

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Screenshot 2.2 The spreadsheet for constructing the efficient frontier

First, let us calculate the minimum variance portfolio. To do this, click on cell N19, which is the one containing the portfolio variance formula. Then click on the Data tab and then on Solver.9 A window will appear which should be completed as in Screenshot 2.3. So we want to minimise cell $N$19 by changing the weights $N$12:$N$14 subject to the constraint that the weights sum to one ($N$15 = 1). Then click Solve. Solver will tell you it has found a solution, so click OK again.

Screenshot 2.3 Completing the Solver window

Note that strictly it is not necessary to use Solver to evaluate this problem when no additional constraints are placed, but if we want to incorporate non-negativity or other constraints on the weights, we could not calculate the weights analytically and Solver would have to be used. The weights in cells N12 to N14 automatically update, as do the portfolio summary statistics in N18 to N20. So the weights that minimise the 139

portfolio variance are with no allocation to Ford, 37% in General Electric and 63% in Microsoft. This achieves a variance of 41 (standard deviation of 6.41%) per month and an average return of 0.33% per month. So we now have one point on the efficient frontier (the one on the far left), and we repeat this procedure to obtain other points on the frontier. We set a target variance and find the weights that maximise the return subject to this variance. In cells N25 to N40, we specify the target standard deviations from 6.5 to 17, increasing in units of 0.5. These figures are somewhat arbitrary, but as a rule of thumb, to get a nice looking frontier, we should have the maximum standard deviation (17) about three times the minimum (6.5). We know not to set any number less than 6.41 since this was the minimum possible standard deviation with these three stocks. We click on the cell N18 and then select Solver again from the Data tab. Then we use all of the entries as before, except that we want to choose Max (to maximise the return subject to a standard deviation constraint) and then add an additional constraint that $N$20 = $N$25, so that the portfolio standard deviation will be equal to the value we want, which is 6.5 in cell N25. Click Solve and the new solution will be found. The weights are now 4% in Ford, 30% in GE, and 66% in Microsoft, giving a mean return of 0.38% and a standard deviation of 6.5(%). Repeat this again for the other standard deviation values from 6.5 through to 17, each time noting the corresponding mean value (and if you wish, also noting the weights). You will see that if you try to find a portfolio with a standard deviation of 17.5, Solver will not be able to find a solution because there are no combinations of the three stocks that will give such a high value. In fact, the upper left point on the efficient frontier will be the maximum return portfolio which will always be 100% invested in the stock with the highest return (in this case Ford). We can now plot the efficient frontier – i.e., the mean return on the yaxis against the standard deviation on the x-axis. If we also want the lower part of the mean–variance opportunity set (the part where the curve folds back on itself at the bottom), we repeat the procedure above – i.e., targeting the standard deviation of 6.5, 7, …, but this time we minimise the return rather than maximising it. The minimum return is 0.24 when the portfolio is 100% invested in GE. The plot will appear as in screenshot 2.4. The line is somewhat wiggly, but this arises because the points are insufficiently close together. If we had used standard deviations from 6.5 to 17 in increments of 0.2, say, rather than 0.5 then the plot would have been much smoother. 140

Screenshot 2.4 A plot of the completed efficient frontier

The final step in the process is to superimpose the capital market line (CML) onto the plot. To do this, we need to find the tangency point, which will be the point at which the Sharpe ratio of the portfolio is maximised. So first we need to calculate the average of the T-bill series (dividing it by twelve to get the monthly rate for comparability with the stock returns, which are monthly), putting this in cell N55. We then calculate the risk premium in N56, which is the risky portfolio return from N18 less the risk-free rate in N56. Finally, the Sharpe ratio in N57 is the risk premium from N56 divided by the portfolio standard deviation (N20). We then get Solver to maximise the value of N57 subject to the weights adding to one (no other constraints are needed). The tangency point has a mean return of exactly 1% per month (by coincidence), standard deviation 12.41% and weights of 66%, 0% and 34% in Ford, GE and Microsoft, respectively. We then need a set of points on the CML to plot − one will be the point on the y-axis where the risk is zero and the return is the average risk-free rate (0.14% per month). Another will be the tangency point we just derived. To get the others, recall that the CML is a straight line with equation return = Rf + Sharpe ratio × std dev. So all we need to do is to use a run of standard deviations and then calculate the corresponding returns − we know that Rf = 0.14 and the Sharpe ratio = 0.0694. The minimum variance opportunity set and the CML on the same graph will appear as in screenshot 2.5. 141

Screenshot 2.5 The capital market line and efficient frontier

KEY CONCEPTS The key terms to be able to define and explain from this chapter are cardinal, ordinal and nominal numbers financial econometrics time-series data panel data continuous data real and nominal series quantiles arithmetic progression mean skewness covariance population present value internal rate of return geometric mean continuously compounded returns cross-sectional data pooled data discrete data deflator 142

coefficient of variation geometric progression variance kurtosis correlation sample future value

SELF-STUDY QUESTIONS 1. Expand the parentheses as far as possible for the following expressions (a) E(ax + by) for x,y variables and a,b scalars (b) E(axy) for x,y independent variables and a a scalar (c) E(axy) for x,y correlated variables and a a scalar 2. (a) Explain the difference between a pdf and a cdf (b) What shapes are the pdf and cdf for a normally distributed random variable? 3. What is the central limit theorem and why is it important in statistics? 4. Explain the differences between the mean, mode and median. Which is the most useful measure of an average and why? 5. Which is a more useful measure of central tendency for stock returns − the arithmetic mean or the geometric mean? Explain your answer. 6. The covariance between two variables is 0.99. Are they strongly related? Explain your answer. 7. Explain the differences between the following pairs of terms (a) Continuous and discrete data (b) Ordinal and nominal data (c) Time-series and panel data (d) Noisy and clean data (e) Simple and continuously compounded returns (f) Nominal and real series 143

(g) Bayesian and classical statistics 8. Present and explain a problem that can be approached using a time-series regression, another one using cross-sectional regression, and another using panel data. 9. What are the key features of asset return time-series? 10. The following table gives annual, end of year prices of a bond and of the consumer prices index Year 2011 2012 2013 2014 2015 2016 2017 2018

Bond value CPI value 36.9 108.0 39.8 110.3 42.4 113.6 38.1 116.1 36.4 118.4 39.2 120.9 44.6 123.2 45.1 125.4

(a) Calculate the simple returns (b) Calculate the continuously compounded returns (c) Calculate the prices of the bond each year in 2018 terms (d) Calculate the real returns 11. Start with part of the formula for calculating an effective interest rate from a nominal one

Assume that the interest rate, r is 10%, and use T =1, 2, 5, 10, 20, 50, 100, 200, 500, 1000, 5000 to calculate the corresponding value of X in each case. Produce a graph where you plot X on the y-axis and T on the x-axis. What do you notice happening as X is increasing and can you see what would happen as X rises to infinity? 12. Suppose that I place £1000 today in a savings account that pays 3% interest once per year and I make no additions to the account. (a) How many years would it take for me to double my money? (b) If instead I switch to a better account that pays 5% per year, 144

how many years would it take? (c) If I want to triple the money with a 5% interest rate, how long would it take? (d) If the 5% was paid continuously, how many years would it take to double the money? 13. A saver has a choice of two accounts: one paying 12%, compounded annually, and one paying 11%, compounded monthly. Which should he or she choose and why? 14. (a) Starting now, if I save £200 per month, is it possible that I will become a millionaire in my lifetime assuming a 5% annual growth rate in my investments, compounded annually? (b) Suppose that I manage instead to save £500 per month, is it possible that I will become a millionaire? (c) Suppose now that I can save £500 per month but the effects of inflation imply that the real rate of growth in the value of my investments is only 2%. Will I become a millionaire in today’s money terms?

1 2

3 4

5

A more precise and complete definition of the median is surprisingly complex, but is not necessary for our purposes. N is used here to denote the number of observations – i.e., the sample size, which is denoted as T in later chapters for consistency with the standard approach in the time-series literature. Note that there are several slightly different formulae that can be used for calculating quartiles, each of which may provide slightly different answers. Of course, we could also define the positive semi-variance and positive semi-standard deviation where only observations such that are included in the sum. There are a number of ways to calculate skewness (and kurtosis); the one given in the formula is sometimes known as the moment coefficient of skewness, but it could also be measured using the standardised difference between the mean and the median, or by using the quartiles of the data. Unfortunately, this implies that different software packages will give slightly different values for the skewness and kurtosis coefficients. Also, some packages make a ‘degrees of freedom correction’ as we do in the equations here, while others do not, so that the divisor in such cases would be N rather than N − 1 in the equations.

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6 7

8 9

Strictly, the variance is the second moment, not the standard deviation. Discretely measured data do not necessarily have to be integers. For example, until they became ‘decimalised’, many financial asset prices were quoted to the nearest 1/16 or 1/32 of a dollar. Note that w′· 1N will be 1 × 1 – i.e., a scalar. Note that you may have to load the Solver Add-In: see the online Microsoft support for how to do this for your version of Excel and platform.

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3 A Brief Overview of the Classical Linear Regression Model

LEARNING OUTCOMES In this chapter, you will learn how to Derive the OLS formulae for estimating parameters and their standard errors Explain the desirable properties that a good estimator should have Discuss the factors that affect the sizes of standard errors Test hypotheses using the test of significance and confidence interval approaches Interpret p-values

3.1 What is a Regression Model? Regression analysis is almost certainly the most important tool at the econometrician’s disposal. But what is regression analysis? In very general terms, regression is concerned with describing and evaluating the relationship between a given variable and one or more other variables. More specifically, regression is an attempt to explain movements in a variable by reference to movements in one or more other variables. To make this more concrete, denote the variable whose movements the regression seeks to explain by y and the variables which are used to explain those variations by x1, x2, …, xk. Hence, in this relatively simple setup, it would be said that variations in k variables (the xs) cause changes in some other variable, y. This chapter will be limited to the case where the 147

model seeks to explain changes in only one variable y (although this restriction will be removed in Chapter 7). There are various completely interchangeable names for y and the xs, and all of these terms will be used synonymously in this book (see Box 3.1). BOX 3.1 Names for y and xs in regression models Names for y Dependent variable Regressand Effect variable Explained variable

Names for the xs Independent variables Regressors Causal variables Explanatory variables

3.2 Regression versus Correlation As discussed in Chapter 2, the correlation between two variables measures the degree of linear association between them. If it is stated that y and x are correlated, it means that y and x are being treated in a completely symmetrical way. Thus, it is not implied that changes in x cause changes in y, or indeed that changes in y cause changes in x. Rather, it is simply stated that there is evidence for a linear relationship between the two variables, and that movements in the two are on average related to an extent given by the correlation coefficient. In regression, the dependent variable (y) and the independent variable(s) (xs) are treated very differently. The y variable is assumed to be random or ‘stochastic’ in some way, i.e., to have a probability distribution. The x variables are, however, assumed to have fixed (‘non-stochastic’) values in repeated samples.1 Regression as a tool is more flexible and more powerful than correlation.

3.3 Simple Regression For simplicity, suppose for now that it is believed that y depends on only one x variable. Again, this is of course a severely restricted case, but the case of more explanatory variables will be considered in Chapter 4. Three examples of the kind of relationship that may be of interest include

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How asset returns vary with their level of market risk Measuring the long-term relationship between stock prices and dividends Constructing an optimal hedge ratio Suppose that a researcher has some idea that there should be a relationship between two variables y and x, and that financial theory suggests that an increase in x will lead to an increase in y. A sensible first stage to testing whether there is indeed an association between the variables would be to form a scatter plot of them. Suppose that the outcome of this plot is Figure 3.1.

Figure 3.1 Scatter plot of two variables, y and x

In this case, it appears that there is an approximate positive linear relationship between x and y which means that increases in x are usually accompanied by increases in y, and that the relationship between them can be described approximately by a straight line. It would be possible to draw by hand onto the graph a line that appears to fit the data. The intercept and slope of the line fitted by eye could then be measured from the graph. However, in practice such a method is likely to be laborious and inaccurate. It would therefore be of interest to determine to what extent this relationship can be described by an equation that can be estimated using a defined procedure. It is possible to use the general equation for a straight 149

line (3.1) to get the line that best ‘fits’ the data. The researcher would then be seeking to find the values of the parameters or coefficients, α and β, which would place the line as close as possible to all of the data points taken together. However, this equation (y = α + βx) is an exact one. Assuming that this equation is appropriate, if the values of α and β had been calculated, then given a value of x, it would be possible to determine with certainty what the value of y would be. Imagine – a model which says with complete certainty what the value of one variable will be given any value of the other! Clearly this model is not realistic. Statistically, it would correspond to the case where the model fitted the data perfectly – that is, all of the data points lay exactly on a straight line. To make the model more realistic, a random disturbance term, denoted by u, is added to the equation, thus (3.2) where the subscript t (= 1, 2, 3, …) denotes the observation number. The disturbance term can capture a number of features (see Box 3.2). BOX 3.2 Reasons for the inclusion of the disturbance term Even in the general case where there is more than one explanatory variable, some determinants of yt will always in practice be omitted from the model. This might, for example, arise because the number of influences on y is too large to place in a single model, or because some determinants of y may be unobservable or not measurable. There may be errors in the way that y is measured which cannot be modelled. There are bound to be random outside influences on y that again cannot be modelled. For example, a terrorist attack, a hurricane or a computer failure could all affect financial asset returns in a way that cannot be captured in a model and cannot be forecast reliably. Similarly, many researchers would argue that human behaviour 150

has an inherent randomness and unpredictability!

So how are the appropriate values of α and β determined? α and β are chosen so that the (vertical) distances from the data points to the fitted lines are minimised (so that the line fits the data as closely as possible). The parameters are thus chosen to minimise collectively the (vertical) distances from the data points to the fitted line. This could be done by ‘eye-balling’ the data and, for each set of variables y and x, one could form a scatter plot and draw on a line that looks as if it fits the data well by hand, as in Figure 3.2.

Figure 3.2 Scatter plot of two variables with a line of best fit chosen by

eye Note that the vertical distances are usually minimised rather than the horizontal distances or those taken perpendicular to the line. This arises as a result of the assumption that x is fixed in repeated samples, so that the problem becomes one of determining the appropriate model for y given (or conditional upon) the observed values of x. This ‘eye-balling’ procedure may be acceptable if only indicative results are required, but of course this method, as well as being tedious, is likely to be imprecise. The most common method used to fit a line to the data is known as ordinary least squares (OLS). This approach forms the workhorse of econometric model estimation, and will be discussed in 151

detail in this and subsequent chapters. Two alternative estimation methods (for determining the appropriate values of the coefficients α and β) are the method of moments and the method of maximum likelihood. A generalised version of the method of moments, due to Hansen (1982), is popular, but beyond the scope of this book. The method of maximum likelihood is also widely employed, and will be discussed in detail in Chapter 9. Suppose now, for ease of exposition, that the sample of data contains only five observations. The method of OLS entails taking each vertical distance from the point to the line, squaring it and then minimising the total sum of the areas of squares (hence ‘least squares’), as shown in Figure 3.3. This can be viewed as equivalent to minimising the sum of the areas of the squares drawn from the points to the line.

Figure 3.3 Method of OLS fitting a line to the data by minimising the

sum of squared residuals Tightening up the notation, let yt denote the actual data point for observation t and let denote the fitted value from the regression line – in other words, for the given value of x of this observation t, is the value for y which the model would have predicted. Note that a hat (ˆ) over a variable or parameter is used to denote a value estimated by a model. Finally, let denote the residual, which is the difference between the actual value of y and the value fitted by the model for this data point – i.e., This is shown for just one observation t in Figure 3.4.

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Figure 3.4 Plot of a single observation, together with the line of best fit,

the residual and the fitted value What is done is to minimise the sum of the The reason that the sum of the squared distances is minimised rather than, for example, finding the sum of that is as close to zero as possible, is that in the latter case some points will lie above the line while others lie below it. Then, when the sum to be made as close to zero as possible is formed, the points above the line would count as positive values, while those below would count as negatives. So these distances will in large part cancel each other out, which would mean that one could fit virtually any line to the data, so long as the sum of the distances of the points above the line and the sum of the distances of the points below the line were the same. In that case, there would not be a unique solution for the estimated coefficients. In fact, any fitted line that goes through the mean of the observations (i.e., ) would set the sum of the to zero. However, taking the squared distances ensures that all deviations that enter the calculation are positive and therefore do not cancel out. So minimising the sum of squared distances is given by minimising or minimising

This sum is known as the residual sum of squares (RSS) or the sum of squared residuals. But what is Again, it is the difference between the 153

actual point and the line, So minimising is equivalent to minimising Letting and denote the values of α and β selected by minimising the RSS, respectively, the equation for the fitted line is given by Now let L denote the RSS, which is also known as a loss function. Take the summation over all of the observations, i.e., from t = 1 to T, where T is the number of observations (3.3) L is minimised with respect to (w.r.t.) and to find the values of α and β which minimise the residual sum of squares to give the line that is closest to the data. So L is differentiated w.r.t. and setting the first derivatives to zero. A derivation of the OLS estimator is given in Appendix 3.1 to this chapter. The coefficient estimators for the slope and the intercept are given by (3.4) (3.5) Equations (3.4) and (3.5) state that, given only the sets of observations xt and yt, it is always possible to calculate the values of the two parameters, and that best fit the set of data. Equation (3.4) is the easiest formula to use to calculate the slope estimate, but the formula can also be written, more intuitively, as (3.6) which is equivalent to the sample covariance between x and y divided by the sample variance of x. To reiterate, this method of finding the optimum is known as OLS. By construction it finds the parameter values that best fit the sample data: any other parameter values would lead to a worse fit and a higher RSS. This is illustrated in Figure 3.5, which shows how the RSS varies with different values of β and gives the lowest value. As an exercise, you could set up a spreadsheet that calculates the RSS from equation (3.5) and the sample 154

data in the following Example 3.3, produce a plot similar to Figure 3.5 and demonstrate that any other values of β will give a higher RSS.

Figure 3.5 How RSS varies with different values of β

It is also worth noting that it is obvious from the equation for that the regression line will go through the mean of the observations – i.e., that the point lies on the regression line. EXAMPLE 3.1 Suppose that some data have been collected on the excess returns on a fund manager’s portfolio (‘fund XXX’) together with the excess returns on a market index, as shown in Table 3.1. Table 3.1 Sample data on fund XXX to motivate OLS estimation

The fund manager has some intuition that the beta (in the CAPM framework) on this fund is positive, and she therefore wants to find whether there appears to be a relationship between x and y given the data. Again, the first stage could be to form a scatter plot of the two 155

variables (Figure 3.6).

Figure 3.6 Scatter plot of excess returns on fund XXX versus excess

returns on the market portfolio Clearly, there appears to be a positive, approximately linear relationship between x and y, although there is not much data on which to base this conclusion! Plugging the five observations in to make up the formulae given in equations (3.5) and (3.4) would lead to the estimates and The fitted line would be written as (3.7) where xt is the excess return of the market portfolio over the risk-free rate (i.e., rm – rf), also known as the market risk premium.

3.3.1 What are

and

Used For?

This question is probably best answered by posing another question. If an analyst tells you that she expects the market to yield a return 20% higher than the risk-free rate next year, what would you expect the return on fund XXX to be? The expected value of y = ‘–1.74 + 1.64 × value of x’, so plug x = 20 into (3.7)

156

(3.8) Thus, for a given expected market risk premium of 20%, and given its riskiness, fund XXX would be expected to earn an excess over the riskfree rate of approximately 31%. In this setup, the regression beta is also the CAPM beta, so that fund XXX has an estimated beta of 1.64, suggesting that the fund is rather risky. In this case, the residual sum of squares reaches its minimum value of 30.33 with these OLS coefficient values. Although it may be obvious, it is worth stating that it is not advisable to conduct a regression analysis using only five observations! Thus the results presented here can be considered indicative and for illustration of the technique only. Some further discussions on appropriate sample sizes for regression analysis are given in Chapter 5. The coefficient estimate of 1.64 for β is interpreted as saying that, ‘if x increases by 1 unit, y will be expected, everything else being equal, to increase by 1.64 units’. Of course, if had been negative, a rise in x would on average cause a fall in y. the intercept coefficient estimate, is interpreted as the value that would be taken by the dependent variable y if the independent variable x took a value of zero. ‘Units’ here refer to the units of measurement of xt and yt. So, for example, suppose that x is measured in per cent and y is measured in thousands of US dollars. Then it would be said that if x rises by 1%, y will be expected to rise on average by $1.64 thousand (or $1,640). Mathematically, we can interpret the slope coefficient in a regression model as being the derivative of the dependent variable with respect to the independent variable, i.e., In cases where there is more than one independent variable (as we will meet in Chapter 4), then the coefficients can be interpreted as partial derivatives of the dependent variable with respect to each independent variable, Note that changing the scale of y or x will make no difference to the overall results since the coefficient estimates will change by an off-setting factor to leave the overall relationship between y and x unchanged (see Gujarati, 2003, pp. 169–73 for a proof). Thus, if the units of measurement of y were hundreds of dollars instead of thousands, and everything else remains unchanged, the slope coefficient estimate would be 16.4, so that a 1% increase in x would lead to an increase in y of $16.4 hundreds (or $1,640) as before. All other properties of the OLS estimator discussed below are also invariant to changes in the scaling of the data. 157

A word of caution is, however, in order concerning the reliability of estimates of the constant term. Although the strict interpretation of the intercept is indeed as stated above, in practice, it is often the case that there are no values of x close to zero in the sample. In such instances, estimates of the value of the intercept will be unreliable. For example, consider Figure 3.7, which demonstrates a situation where no points are close to the y-axis.

Figure 3.7 No observations close to the y-axis

In such cases, one could not expect to obtain robust estimates of the value of y when x is zero as all of the information in the sample pertains to the case where x is considerably larger than zero. A similar caution should be exercised when producing predictions for y using values of x that are a long way outside the range of values in the sample. In Example 3.1, x takes values between 7% and 23% in the available data. So, it would not be advisable to use this model to determine the expected excess return on the fund if the expected excess return on the market were, say 1% or 30%, or –5% (i.e., the market was expected to fall).

3.4 Some Further Terminology 3.4.1 The Data Generating Process, the Population 158

Regression Function and the Sample Regression Function The population regression function (PRF) is a description of the model that is thought to be generating the actual data and it represents the true relationship between the variables. The population regression function is also known as the data generating process (DGP). The PRF embodies the true values of α and β, and is expressed as (3.9) Note that there is a disturbance term in this equation, so that even if one had at one’s disposal the entire population of observations on x and y, it would still in general not be possible to obtain a perfect fit of the line to the data. In some textbooks, a distinction is drawn between the PRF (the underlying true relationship between y and x) and the DGP (the process describing the way that the actual observations on y come about), although in this book, the two terms will be used synonymously. The sample regression function (SRF) is the relationship that has been estimated using the sample observations, and is often written as (3.10) Notice that there is no error or residual term in equation (3.10); all this equation states is that given a particular value of x, multiplying it by and adding will give the model fitted or expected value for y, denoted It is also possible to write (3.11) Equation (3.11) splits the observed value of y into two components: the fitted value from the model, and a residual term. The SRF is used to infer likely values of the PRF. That is, the estimates and are constructed, for the sample of data at hand, but what is really of interest is the true relationship between x and y – in other words, the PRF is what is really wanted, but all that is ever available is the SRF. However, what can be said is how likely it is, given the figures calculated for and that the corresponding population parameters take on certain values.

159

3.4.2 Linearity and Possible Forms for the Regression Function In order to use OLS, a model that is linear is required. This means that, in the simple bivariate case, the relationship between x and y must be capable of being expressed diagramatically using a straight line. More specifically, the model must be linear in the parameters (α and β), but it does not necessarily have to be linear in the variables (y and x). By ‘linear in the parameters’, it is meant that the parameters are not multiplied together, divided, squared or cubed, etc. Models that are not linear in the variables can often be made to take a linear form by applying a suitable transformation or manipulation. For example, consider the following exponential regression model (3.12) Taking logarithms of both sides, applying the laws of logs and rearranging the RHS (3.13) where A and β are parameters to be estimated. Now let α = ln(A), yt = ln Yt and xt = lnXt (3.14) This is known as an exponential regression model since Y varies according to some exponent (power) function of X. In fact, when a regression equation is expressed in ‘double logarithmic form’, which means that both the dependent and the independent variables are natural logarithms, the coefficient estimates are interpreted as elasticities (strictly, they are unit changes on a logarithmic scale). Elasticities are useful since they are unit-free – that is, they are not a function of the units of measurement of either the dependent or the independent variable. Mathematically, as stated above, the slope parameter estimate in a linear regression (not in logarithmic form) can be interpreted as a derivative of y with respect to x. This is sometimes known as a marginal propensity. Elasticities can also be calculated in this context by taking

160

– in other words, multiplying the derivative (i.e., the slope in the regression) by the value of x at some point (call this x0) and dividing by the corresponding value of y, y0. We can see from the left-hand side of the expression why this is unit for both x and y have effectively been cancelled out. From the right-hand side, we can see that the elasticity is measuring the proportional change ratio, which is the amount that y changes, dy, as a proportion of its actual value, y0, divided by the amount that x changes, dx, as a proportion of its actual value, x0. Thus a coefficient estimate of 1.2 for in equation (3.13) or (3.14) is interpreted as stating that ‘a rise in X of 1% will lead on average, everything else being equal, to a rise in Y of 1.2%’. Conversely, for y and x in levels (e.g., equation (3.9)) rather than logarithmic form, the coefficients denote unit changes as described above. Similarly, if theory suggests that x should be inversely related to y according to a model of the form (3.15) the regression can be estimated using OLS by setting

and regressing y on a constant and z. Clearly, then, a surprisingly varied array of models can be estimated using OLS by making suitable transformations to the variables. On the other hand, some models are intrinsically non-linear, e.g. (3.16) Such models cannot be estimated using OLS, but might be estimable using a nonlinear estimation method (see Chapter 9).

3.4.3 Estimator or Estimate? Estimators are the formulae used to calculate the coefficients – for example, the expressions given in equation (3.4) and (3.5) above, while the 161

estimates, on the other hand, are the actual numerical values for the coefficients that are obtained from the sample.

3.5 The Assumptions Underlying the Classical Linear Regression Model The model yt = α + βxt + ut that has been derived above, together with the assumptions listed below, is known as the classical linear regression model (CLRM). Data for xt are observable, but since yt also depends on ut, it is necessary to be specific about how the ut are generated. The set of assumptions shown in Box 3.3 are usually made concerning the ut s, the unobservable error or disturbance terms. Note that no assumptions are made concerning their observable counterparts, the estimated model’s residuals. As long as assumption (1) holds, assumption (4) can be equivalently written E(xt ut) = 0. Both formulations imply that the regressor is orthogonal to (i.e., unrelated to) the error term. An alternative assumption to (4), which is slightly stronger, is that the xt are non-stochastic or fixed in repeated samples. This means that there is no sampling variation in xt, and that its value is determined outside the model. BOX 3.3 Assumptions concerning disturbance terms and their interpretation Technical notation (1) E(ut) = 0

Interpretation The errors have zero mean

(2) var(ut) = σ2 < ∞

The variance of the errors is constant and finite over all values of xt

(3) cov(ui, uj) = 0

The errors are linearly independent of one another There is no relationship between the error and corresponding x variate – i.e., that ut is normally distributed.

(4) cov(ut, xt) = 0 (5) ut ~ N(0, σ2)

A fifth assumption is required to make valid inferences about the population parameters (the actual α and β) from the sample parameters ( 162

and ) estimated using a finite amount of data, namely that the disturbances follow a normal distribution.

3.6 Properties of the OLS Estimator If assumptions (1)–(4) hold, then the estimators and determined by OLS will have a number of desirable properties, and are known as best linear unbiased estimators (BLUE). What does this acronym stand for? ‘Estimator’ – and are estimators of the true value of α and β ‘Linear’ – and are linear estimators – that means that the formulae for and are linear combinations of the random variables (in this case, y) ‘Unbiased’ – on average, the actual values of and will be equal to their true values ‘Best’ – means that the OLS estimator has minimum variance among the class of linear unbiased estimators; the Gauss–Markov theorem proves that the OLS estimator is best by examining an arbitrary alternative linear unbiased estimator and showing in all cases that it must have a variance no smaller than the OLS estimator Under assumptions (1)–(4) listed above, the OLS estimator can be shown to have the desirable properties that it is consistent, unbiased and efficient. Unbiasedness and efficiency have already been discussed above, and consistency is an additional desirable property. These three characteristics will now be discussed in turn.

3.6.1 Consistency The least squares estimators and are consistent. One way to state this algebraically for (with the obvious modifications made for ) is (3.17) This is a technical way of stating that the probability (Pr) that is more than some arbitrary fixed distance δ away from its true value tends to zero as the sample size tends to infinity, for all positive values of δ. Thus β is the probability limit of In the limit (i.e., for an infinite number of observations), the probability of the estimator being different from the true value is zero. That is, the estimates will converge to their true values as the 163

sample size increases to infinity. Consistency is thus a large sample, or asymptotic property. If an estimator is inconsistent, then even if we had an infinite amount of data, we could not be sure that the estimated value of a parameter will be close to its true value. So consistency is sometimes argued to be the most important property of an estimator. The assumptions that E(xtut) = 0 and E(ut) = 0 are sufficient to derive the consistency of the OLS estimator.

3.6.2 Unbiasedness The least squares estimates of and

are unbiased. That is (3.18)

and (3.19) Thus, on average, the estimated values for the coefficients will be equal to their true values. That is, there is no systematic overestimation or underestimation of the true coefficients. To prove this also requires the assumption that cov(ut, xt) = 0. Clearly, unbiasedness is a stronger condition than consistency, since it holds for small as well as large samples (i.e., for all sample sizes). An estimator that is consistent may still be biased for small samples, but are all unbiased estimators also consistent? The answer is in fact ‘no’. An unbiased estimator will also be consistent if its variance falls as the sample size increases.

3.6.3 Efficiency An estimator of a parameter β is said to be efficient if no other estimator has a smaller variance. Broadly, if the estimator is efficient, it will be minimising the probability that it is a long way off from the true value of β or, in simpler terms, the variation in the parameter estimate from one sample within the population to another would be minimised. In other words, if the estimator is ‘best’, the uncertainty associated with estimation will be minimised for the class of linear unbiased estimators. A technical way to state this would be to say that an efficient estimator would have a probability distribution that is narrowly dispersed around the true value.

164

3.6.4 More on Unbiasedness and Efficiency As stated above, the Gauss–Markov theorem shows that the OLS estimator has the least variance among the class of linear unbiased estimators. It is worth exploring these concepts in a little more detail to try to get a better grip on their meaning and implication. It would be possible to find an estimator with a lower variance than the OLS estimator, but it would not be linear and unbiased. As obvious example would be a fixed estimator, e.g., so in other words, whatever the data say, we fix the slope estimate at 2. This estimator would clearly have a lower variance than the OLS estimator – in fact, it would have a variance of zero since there would be no change in the parameter estimate from one sample to another since it is always 2 – but it would clearly be biased and inconsistent, as we increase the sample size, there would be no convergence upon the true population value of β, and the error between our estimated value of 2 and the true value would always be in the same direction and thus biased. More generally, other data-dependent (non-OLS) estimators could be used but these would either be non-linear or biased or have a higher variance than the OLS estimator. Thus, there is often a trade-off between bias and variance, so that to improve one means worsening the other. The situation is illustrated in Figure 3.8, which plots the distributions of two different estimators. These both display the ranges of estimates for the slope parameter that might arise when selecting different samples from within the population. The distribution which has at its centre represents the OLS estimator – this has the true value (β) as the most commonly estimated value in the centre, and is thus unbiased, but it also has a bigger variance than the other estimator, as it is flatter and with more of the distribution in the tails and less in the centre. On the other hand, the alternative estimator, which is represented by the distribution to the right of Figure 3.8, is much more focused on its mean value (which I have called ) – this estimator clearly has a lower variance but is biased since its centre is not on the true value of β. Up to a certain point, bias is usually considered a more serious problem than variance and hence the widespread use of the OLS estimator as the core of econometric model– building.

165

Figure 3.8 The bias versus variance trade-off when selecting between

estimators

3.7 Precision and Standard Errors Any set of regression estimates and are specific to the sample used in their estimation. In other words, if a different sample of data was selected from within the population, the data points (the xt and yt) will be different, leading to different values of the OLS estimates. Recall that the OLS estimators ( and ) are given by equation (3.4) and (3.5). It would be desirable to have an idea of how ‘good’ these estimates of α and β are in the sense of having some measure of the reliability or precision of the estimators ( and ). It is thus useful to know whether one can have confidence in the estimates, and whether they are likely to vary much from one sample to another sample within the given population. An idea of the sampling variability and hence of the precision of the estimates can be calculated using only the sample of data available. This estimate of precision is given by its standard error. Given assumptions (1)–(4) above, valid estimators of the standard errors can be shown to be given by

(3.20)

(3.21) where s is the estimated standard deviation of the residuals (see below). These formulae are derived in Appendix 3.1 to this chapter. 166

It is worth noting that the standard errors give only a general indication of the likely accuracy of the regression parameters. They do not show how accurate a particular set of coefficient estimates is. If the standard errors are small, it shows that the coefficients are likely to be precise on average, not how precise they are for this particular sample. Thus standard errors give a measure of the degree of uncertainty in the estimated values for the coefficients. It can be seen that they are a function of the actual observations on the explanatory variable, x, the sample size, T, and another term, s. The last of these is an estimate of the variance of the disturbance term. The actual variance of the disturbance term is usually denoted by σ2. How can an estimate of σ2 be obtained?

3.7.1 Estimating the Variance of the Error Term (σ2) From elementary statistics, the variance of a random variable ut is given by (3.22) Assumption (1) of the CLRM was that the expected or average value of the errors is zero. Under this assumption, equation (3.22) above reduces to (3.23) So what is required is an estimate of the average value of be calculated as

which could

(3.24) Unfortunately equation (3.24) is not workable since ut is a series of population disturbances, which is not observable. Thus the sample counterpart to ut, which is is used (3.25) But this estimator is a biased estimator of σ2. An unbiased estimator, s2, would be given by equation (3.26) instead of equation (3.25) (3.26) 167

where is the residual sum of squares, so that the quantity of relevance for the standard error formulae is the square root of equation (3.26) (3.27) s is also known as the standard error of the regression or the standard error of the estimate. It is sometimes used as a broad measure of the fit of the regression equation. Everything else being equal, the smaller this quantity is, the closer is the fit of the line to the actual data.

3.7.2 Some Comments on the Standard Error Estimators It is possible, of course, to derive the formulae for the standard errors of the coefficient estimates from first principles using some algebra, and this is left to Appendix 3.1 to this chapter. Some general intuition is now given as to why the formulae for the standard errors given by equations (3.20) and (3.21) contain the terms that they do and in the form that they do. The presentation offered in Box 3.4 loosely follows that of Hill, Griffiths and Judge (1997), which is the clearest that this author has seen. BOX 3.4 Standard error estimators (1) The larger the sample size, T, the smaller will be the coefficient standard errors. T appears explicitly in and implicitly in T appears implicitly since the sum is from t = 1 to T. The reason for this is simply that, at least for now, it is assumed that every observation on a series represents a piece of useful information which can be used to help determine the coefficient estimates. So the larger the size of the sample, the more information will have been used in estimation of the parameters, and hence the more confidence will be placed in those estimates. (2) Both and depend on s2 (or s). Recall from above that s2 is the estimate of the error variance. The larger this quantity is, the more dispersed are the residuals, and so the greater is the uncertainty in the model. If s2 is large, the data points are collectively a long way away from the line. (3) The sum of the squares of the xt about their mean appears in 168

both formulae – since appears in the denominators. The larger the sum of squares, the smaller the coefficient variances. Consider what happens if is small or large, as shown in Figures 3.9 and 3.10, respectively. In Figure 3.9, the data are close together so that is small. In this first case, it is more difficult to determine with any degree of certainty exactly where the line should be. On the other hand, in Figure 3.10, the points are widely dispersed across a long section of the line, so that one could hold more confidence in the estimates in this case.

Figure 3.9 Effect on the standard errors of the coefficient estimates

when

are narrowly dispersed

169

Figure 3.10 Effect on the standard errors of the coefficient estimates

when

are widely dispersed

(4) The term affects only the intercept standard error and not the slope standard error. The reason is that measures how far the points are away from the y-axis. Consider Figures 3.11 and 3.12. In Figure 3.11, all of the points are bunched a long way from the y-axis, which makes it more difficult to accurately estimate the point at which the estimated line crosses the y-axis (the intercept). In Figure 3.12, the points collectively are closer to the y-axis and hence it will be easier to determine where the line actually crosses the axis. Note that this intuition will work only in the case where all of the xt are positive!

170

Figure 3.11 Effect on the standard errors of

large

Figure 3.12 Effect on the standard errors of

small

EXAMPLE 3.2 Assume that the following data have been calculated from a regression of y on a single variable x and a constant over twenty-two observations

171

Determine the appropriate values of the coefficient estimates and their standard errors. This question can simply be answered by plugging the appropriate numbers into the formulae given above. The calculations are

The sample regression function would be written as

Now, turning to the standard error calculations, it is necessary to obtain an estimate, s, of the error variance

With the standard errors calculated, the results are written as (3.28) The standard error estimates are usually placed in parentheses under the relevant coefficient estimates.

3.8 An Introduction to Statistical Inference Often, financial theory will suggest that certain coefficients should take on particular values, or values within a given range. It is thus of interest to determine whether the relationships expected from financial theory are upheld by the data to hand or not. Estimates of α and β have been obtained 172

from the sample, but these values are not of any particular interest; the population values that describe the true relationship between the variables would be of more interest, but are never available. Instead, inferences are made concerning the likely population values from the regression parameters that have been estimated from the sample of data to hand. In doing this, the aim is to determine whether the differences between the coefficient estimates that are actually obtained, and expectations arising from financial theory, are a long way from one another in a statistical sense. EXAMPLE 3.3 Suppose the following regression results have been calculated: (3.29) is a single (point) estimate of the unknown population parameter, β. As stated above, the reliability of the point estimate is measured by the coefficient’s standard error. The information from one or more of the sample coefficients and their standard errors can be used to make inferences about the population parameters. So the estimate of the slope coefficient is but it is obvious that this number is likely to vary to some degree from one sample to the next. It might be of interest to answer the question, ‘Is it plausible, given this estimate, that the true population parameter, β, could be 0.5? Is it plausible that β could be 1?’, etc. Answers to these questions can be obtained through hypothesis testing.

3.8.1 Hypothesis Testing: Some Concepts In the hypothesis testing framework, there are always two hypotheses that go together, known as the null hypothesis (denoted H0 or occasionally HN) and the alternative hypothesis (denoted H1 or occasionally HA). The null hypothesis is the statement or the statistical hypothesis that is actually being tested. The alternative hypothesis represents the remaining outcomes of interest. For example, suppose that given the regression results above, it is of interest to test the hypothesis that the true value of β is in fact 0.5. The following notation would be used. 173

This states that the hypothesis that the true but unknown value of β could be 0.5 is being tested against an alternative hypothesis where β is not 0.5. This would be known as a two-sided test, since the outcomes of both β < 0.5 and β > 0.5 are subsumed under the alternative hypothesis. Sometimes, some prior information may be available, suggesting for example that β > 0.5 would be expected rather than β < 0.5. In this case, β < 0.5 is no longer of interest to us, and hence a one-sided test would be conducted:

Here the null hypothesis that the true value of β is 0.5 is being tested against a one-sided alternative that β is more than 0.5. On the other hand, one could envisage a situation where there is prior information that β < 0.5 is expected. For example, suppose that an investment bank bought a piece of new risk management software that is intended to better track the riskiness inherent in its traders’ books and that β is some measure of the risk that previously took the value 0.5. Clearly, it would not make sense to expect the risk to have risen, and so β > 0.5, corresponding to an increase in risk, is not of interest. In this case, the null and alternative hypotheses would be specified as

This prior information should come from the financial theory of the problem under consideration, and not from an examination of the estimated value of the coefficient. Note that there is always an equality under the null hypothesis. So, for example, β < 0.5 would not be specified under the null hypothesis. There are two ways to conduct a hypothesis test: via the test of significance approach or via the confidence interval approach. Both methods centre on a statistical comparison of the estimated value of the coefficient, and its value under the null hypothesis. In very general terms, if the estimated value is a long way away from the hypothesised value, the null hypothesis is likely to be rejected; if the value under the null hypothesis and the estimated value are close to one another, the null hypothesis is less likely to be rejected. For example, consider as 174

above. A hypothesis that the true value of β is 5 is more likely to be rejected than a null hypothesis that the true value of β is 0.5. What is required now is a statistical decision rule that will permit the formal testing of such hypotheses.

3.8.2 The Probability Distribution of the Least Squares Estimators In order to test hypotheses, assumption (5) of the CLRM must be used, namely that ut ~ N(0, σ2) – i.e., that the error term is normally distributed. The normal distribution is a convenient one to use for it involves only two parameters (its mean and variance). This makes the algebra involved in statistical inference considerably simpler than it otherwise would have been. Since yt depends partially on ut, it can be stated that if ut is normally distributed, yt will also be normally distributed. Further, since the least squares estimators are linear combinations of the random variables, i.e., where wt are effectively weights, and since the weighted sum of normal random variables is also normally distributed, it can be said that the coefficient estimates will also be normally distributed. Thus

Will the coefficient estimates still follow a normal distribution if the errors do not follow a normal distribution? Well, briefly, the answer is usually ‘yes’ as a result of the central limit theorem, provided that the other assumptions of the CLRM hold, and the sample size is sufficiently large. The issue of non-normality, how to test for it, and its consequences, will be further discussed in Chapter 5. Standard normal variables can be constructed from and by subtracting the mean and dividing by the square root of the variance

The square roots of the coefficient variances are the standard errors. Unfortunately, the standard errors of the true coefficient values under the PRF are never known – all that is available are their sample counterparts, 2 the calculated standard errors of the coefficient estimates, and Replacing the true values of the standard errors with the sample 175

estimated versions induces another source of uncertainty, and also means that the standardised statistics follow a t-distribution with T – 2 degrees of freedom (defined below) rather than a normal distribution, so

This result is not formally proved here. For a formal proof, see Hill, Griffiths and Judge (1997, pp. 88–90).

3.8.3 A Note on the t and the Normal Distributions The normal distribution pdf was shown in Figure 3.2 with its characteristic ‘bell’ shape and its symmetry around the mean (of zero for a standard normal distribution). Any normal variate can be scaled to have zero mean and unit variance by subtracting its mean and dividing by its standard deviation. There is a specific relationship between the t- and the standard normal distribution, and the t-distribution has another parameter, its degrees of freedom. What does the t-distribution look like? It looks similar to a normal distribution, but with fatter tails, and a smaller peak at the mean, as shown in Figure 3.13.

Figure 3.13 The t-distribution versus the normal

Some examples of the percentiles from the normal and t-distributions taken from the statistical tables are given in Table 3.2 (more critical values for these two distributions are given in Tables A2.1 and A2.2 of Appendix 176

2 at the end of this book). When used in the context of a hypothesis test, these percentiles become critical values. The values presented in Table 3.2 would be those critical values appropriate for a one-sided test of the given significance level. Table 3.2 Critical values from the standard normal versus t-distribution Significance level (%)

N(0,1)

t40

t4

50

0

0

0

5

1.64

1.68

2.13

2.5

1.96

2.02

2.78

0.5

2.57

2.70

4.60

It can be seen that as the number of degrees of freedom for the tdistribution increases from 4 to 40, the critical values fall substantially. In Figure 3.13, this is represented by a gradual increase in the height of the distribution at the centre and a reduction in the fatness of the tails as the number of degrees of freedom increases. In the limit, a t-distribution with an infinite number of degrees of freedom is a standard normal, i.e., t∞ = N(0, 1), so the normal distribution can be viewed as a special case of the t. Putting the limit case, t∞, aside, the critical values for the t-distribution are larger in absolute value than those from the standard normal. This arises from the increased uncertainty associated with the situation where the error variance must be estimated. So now the t-distribution is used, and for a given statistic to constitute the same amount of reliable evidence against the null, it has to be bigger in absolute value than in circumstances where the normal is applicable. As stated above, there are broadly two approaches to testing hypotheses under regression analysis: the test of significance approach and the confidence interval approach. Each of these will now be considered in turn.

3.8.4 The Test of Significance Approach (Box 3.5) Assume the regression equation is given by yt = α + βxt + ut, t = 1, 2, …, T. The steps involved in doing a test of significance are shown in Box 3.5. 177

BOX 3.5 Conducting a test of significance (1) Estimate and in the usual way. (2) Calculate the test statistic. This is given by the formula (3.30) where β* is the value of β under the null hypothesis. The null hypothesis is H0 : β = β* and the alternative hypothesis is H1 : β ≠ β* (for a two-sided test). (3) A tabulated distribution with which to compare the estimated test statistics is required. Test statistics derived in this way can be shown to follow a t-distribution with T – 2 degrees of freedom. (4) Choose a ‘significance level’, often denoted α (not the same as the regression intercept coefficient). It is conventional to use a significance level of 5%. (5) Given a significance level, a rejection region and non-rejection region can be determined. If a 5% significance level is employed, this means that 5% of the total distribution (5% of the area under the curve) will be in the rejection region. That rejection region can either be split in half (for a two-sided test) or it can all fall on one side of the y-axis, as is the case for a one-sided test. For a two-sided test, the 5% rejection region is split equally between the two tails, as shown in Figure 3.14.

178

Figure 3.14 Rejection regions for a two-sided 5% hypothesis test

For a one-sided test, the 5% rejection region is located solely in one tail of the distribution, as shown in Figures 3.15 and 3.16, for a test where the alternative is of the ‘less than’ form, and where the alternative is of the ‘greater than’ form, respectively.

Figure 3.15 Rejection region for a one-sided hypothesis test of the

form H0: β = β*, H1: β < β*

179

Figure 3.16 Rejection region for a one-sided hypothesis test of the

form H0: β = β*, H1: β > β* (6) Use the t-tables to obtain a critical value or values with which to compare the test statistic. The critical value will be that value of x that puts 5% into the rejection region. (7) Finally perform the test. If the test statistic lies in the rejection region then reject the null hypothesis (H0), else do not reject H0. Steps (2)–(7) require further comment. In step (2), the estimated value of β is compared with the value that is subject to test under the null hypothesis, but this difference is ‘normalised’ or scaled by the standard error of the coefficient estimate. The standard error is a measure of how confident one is in the coefficient estimate obtained in the first stage. If a standard error is small, the value of the test statistic will be large relative to the case where the standard error is large. For a small standard error, it would not require the estimated and hypothesised values to be far away from one another for the null hypothesis to be rejected. Dividing by the standard error also ensures that, under the five CLRM assumptions, the test statistic follows a tabulated distribution. In this context, the number of degrees of freedom can be interpreted as the number of pieces of additional information beyond the minimum requirement. If two parameters are estimated (α and β – the intercept and the slope of the line, respectively), a minimum of two observations is required to fit this line to the data. As the number of degrees of freedom increases, the critical values in the tables decrease in absolute terms, since 180

less caution is required and one can be more confident that the results are appropriate. The significance level is also sometimes called the size of the test (note that this is completely different from the size of the sample) and it determines the region where the null hypothesis under test will be rejected or not rejected. Remember that the distributions in Figures 3.14–3.16 are for a random variable. Purely by chance, a random variable will take on extreme values (either large and positive values or large and negative values) occasionally. More specifically, a significance level of 5% means that a result as extreme as this or more extreme would be expected only 5% of the time as a consequence of chance alone. To give one illustration, if the 5% critical value for a one-sided test is 1.68, this implies that the test statistic would be expected to be greater than this only 5% of the time by chance alone. There is nothing magical about the test – all that is done is to specify an arbitrary cutoff value for the test statistic that determines whether the null hypothesis would be rejected or not. It is conventional to use a 5% size of test, but 10% and 1% are also commonly used. However, one potential problem with the use of a fixed (e.g., 5%) size of test is that if the sample size is sufficiently large, any null hypothesis can be rejected. This is particularly worrisome in finance, where tens of thousands of observations or more are often available. What happens is that the standard errors reduce as the sample size increases, thus leading to an increase in the value of all t-test statistics. This problem is frequently overlooked in empirical work, but some econometricians have suggested that a lower size of test (e.g., 1%) should be used for large samples (see, for example, Leamer, 1978, for a discussion of these issues). Note also the use of terminology in connection with hypothesis tests: it is said that the null hypothesis is either rejected or not rejected. It is incorrect to state that if the null hypothesis is not rejected, it is ‘accepted’ (although this error is frequently made in practice), and it is never said that the alternative hypothesis is accepted or rejected. One reason why it is not sensible to say that the null hypothesis is ‘accepted’ is that it is impossible to know whether the null is actually true or not! In any given situation, many null hypotheses will not be rejected. For example, suppose that H0 : β = 0.5 and H0 : β = 1 are separately tested against the relevant two-sided alternatives and neither null is rejected. Clearly then it would not make sense to say that ‘H0 : β = 0.5 is accepted’ and ‘H0 : β = 1 is accepted’, since the true (but unknown) value of β cannot be both 0.5 and 1. So, to summarise, the null hypothesis is either rejected or not rejected on the 181

basis of the available evidence.

3.8.5 The Confidence Interval Approach to Hypothesis Testing (Box 3.6) To give an example of its usage, one might estimate a parameter, say to be 0.93, and a ‘95% confidence interval’ to be (0.77, 1.09). This means that in many repeated samples, 95% of the time, the true value of β will be contained within this interval. Confidence intervals are almost invariably estimated in a two-sided form, although in theory a one-sided interval can be constructed. Constructing a 95% confidence interval is equivalent to using the 5% level in a test of significance. BOX 3.6 Carrying out a hypothesis test using confidence intervals (1) Calculate and as before (2) Choose a significance level, α (again the convention is 5%). This is equivalent to choosing a (1 – α)*100% confidence interval (3) Use the t-tables to find the appropriate critical value, which will again have T–2 degrees of freedom (4) The confidence interval for β is given by

Note that a centre dot (·) is sometimes used instead of a cross (×) to denote when two quantities are multiplied together (5) Perform the test: if the hypothesised value of β (i.e., β*) lies outside the confidence interval, then reject the null hypothesis that β = β*, otherwise do not reject the null.

3.8.6 The Test of Significance and Confidence Interval Approaches Always Give the Same Conclusion Under the test of significance approach, the null hypothesis that β = β* will not be rejected if the test statistic lies within the non-rejection region, i.e., if the following condition holds 182

Rearranging, the null hypothesis would not be rejected if

i.e., one would not reject if

But this is just the rule for non-rejection under the confidence interval approach. So it will always be the case that, for a given significance level, the test of significance and confidence interval approaches will provide the same conclusion by construction. One testing approach is simply an algebraic rearrangement of the other. EXAMPLE 3.4 Given the regression results above (3.31) Using both the test of significance and confidence interval approaches, test the hypothesis that β = 1 against a two-sided alternative. This hypothesis might be of interest, for a unit coefficient on the explanatory variable implies a 1:1 relationship between movements in x and movements in y. The null and alternative hypotheses are, respectively,

The results of the test according to each approach are shown in box 3.7. BOX 3.7 The test of significance and confidence interval approaches compared Test of significance approach

Confidence interval approach

183

Find tcrit = t20;5% = ±2.086

Find tcrit = t20;5% = ±2.086 Do not reject H0 since test statistic lies lies within non-rejection region

Do not reject H0 since 1 within the confidence interval

A couple of comments are in order. First, the critical value from the tdistribution that is required is for twenty degrees of freedom and at the 5% level. This means that 5% of the total distribution will be in the rejection region, and since this is a two-sided test, 2.5% of the distribution is required to be contained in each tail. From the symmetry of the tdistribution around zero, the critical values in the upper and lower tail will be equal in magnitude, but opposite in sign, as shown in Figure 3.17.

Figure 3.17 Critical values and rejection regions for a t20;5%

What if instead the researcher wanted to test H0 : β = 0 or H0 : β = 2? In order to test these hypotheses using the test of significance approach, the test statistic would have to be reconstructed in each case, although the critical value would be the same. On the other hand, no additional work would be required if the confidence interval approach had been adopted, 184

since it effectively permits the testing of an infinite number of hypotheses. So for example, suppose that the researcher wanted to test versus and versus In the first case, the null hypothesis (that β = 0) would not be rejected since 0 lies within the 95% confidence interval. By the same argument, the second null hypothesis (that β = 2) would be rejected since 2 lies outside the estimated confidence interval. On the other hand, note that this book has so far considered only the results under a 5% size of test. In marginal cases (e.g., H0 : β = 1, where the test statistic and critical value are close together), a completely different answer might arise if a different size of test was used. This is where the test of significance approach is preferable to the construction of a confidence interval. For example, suppose that now a 10% size of test is used for the null hypothesis given in Example 3.4. Using the test of significance approach,

as above. The only thing that changes is the critical t-value. At the 10% level (so that 5% of the total distribution is placed in each of the tails for this two-sided test), the required critical value is t20;10% = ±1.725. So now, as the test statistic lies in the rejection region, H0 would be rejected. In order to use a 10% test under the confidence interval approach, the interval itself would have to have been re-estimated since the critical value is embedded in the calculation of the confidence interval. So the test of significance and confidence interval approaches both have their relative merits. The testing of a number of different hypotheses is 185

easier under the confidence interval approach, while a consideration of the effect of the size of the test on the conclusion is easier to address under the test of significance approach. Caution should therefore be used when placing emphasis on or making decisions in the context of marginal cases (i.e., in cases where the null is only just rejected or not rejected). In this situation, the appropriate conclusion to draw is that the results are marginal and that no strong inference can be made one way or the other. A thorough empirical analysis should involve conducting a sensitivity analysis on the results to determine whether using a different size of test alters the conclusions. It is worth stating again that it is conventional to consider sizes of test of 10%, 5% and 1%. If the conclusion (i.e., ‘reject’ or ‘do not reject’) is robust to changes in the size of the test, then one can be more confident that the conclusions are appropriate. If the outcome of the test is qualitatively altered when the size of the test is modified, the conclusion must be that there is no conclusion one way or the other! It is also worth noting that if a given null hypothesis is rejected using a 1% significance level, it will also automatically be rejected at the 5% level, so that there is no need to actually state the latter. Dougherty (1992, p. 100), gives the analogy of a high jumper. If the high jumper can clear 2 metres, it is obvious that the jumper could also clear 1.5 metres. The 1% significance level is a higher hurdle than the 5% significance level. Similarly, if the null is not rejected at the 5% level of significance, it will automatically not be rejected at any stronger level of significance (e.g., 1%). In this case, if the jumper cannot clear 1.5 metres, there is no way she or he will be able to clear 2 metres.

3.8.7 Some More Terminology If the null hypothesis is rejected at the 5% level, it would be said that the result of the test is ‘statistically significant’. If the null hypothesis is not rejected, it would be said that the result of the test is ‘not significant’, or that it is ‘insignificant’. Finally, if the null hypothesis is rejected at the 1% level, the result is termed ‘highly statistically significant’. Note that a statistically significant result may be of no practical significance. For example, if the estimated beta for a stock under a CAPM regression is 1.05, and a null hypothesis that β = 1 is rejected, the result will be statistically significant. But it may be the case that a slightly higher beta will make no difference to an investor’s choice as to whether to buy the stock or not. In that case, one would say that the result of the test was 186

statistically significant but financially or practically insignificant.

3.8.8 Classifying the Errors That Can be Made Using Hypothesis Tests H0 is usually rejected if the test statistic is statistically significant at a chosen significance level. There are two possible errors that could be made: (1) Rejecting H0 when it was really true; this is called a type I error (2) Not rejecting H0 when it was in fact false; this is called a type II error The possible scenarios can be summarised in Table 3.3. The probability of a type I error is just α, the significance level or size of test chosen. To see this, recall what is meant by ‘significance’ at the 5% level: it is only 5% likely that a result as or more extreme as this could have occurred purely by chance. Or, to put this another way, it is only 5% likely that this null would be rejected when it was in fact true. Table 3.3 Classifying hypothesis testing errors and correct conclusions Reality H0 is true Type I error =α

Significant Result of test

H0 is false ✓

(reject H0) Insignificant

✓

Type II error =β

(do not reject H0) Note that there is no chance for a free lunch (i.e., a cost-less gain) here! What happens if the size of the test is reduced (e.g., from a 5% test to a 1% test)? The chances of making a type I error would be reduced …but so would the probability that the null hypothesis would be rejected at all, so increasing the probability of a type II error. The two competing effects of reducing the size of the test are shown in Box 3.8.

187

BOX 3.8 Type I and type II errors

So there always exists, therefore, a direct trade-off between type I and type II errors when choosing a significance level. The only way to reduce the chances of both is to increase the sample size or to select a sample with more variation, thus increasing the amount of information upon which the results of the hypothesis test are based. In practice, up to a certain level, type I errors are usually considered more serious and hence a small size of test is usually chosen (5% or 1% are the most common). The probability of a type I error is the probability of incorrectly rejecting a correct null hypothesis, which is also the size of the test. Another important piece of terminology in this area is the power of a test. The power of a test is defined as the probability of (appropriately) rejecting an incorrect null hypothesis. The power of the test is also equal to one minus the probability of a type II error. In addition to the significance level chosen and the sample size, the power of a statistical test also depends on how ’wrong’ the value proposed under the null hypothesis is compared with the true value. For example, suppose that the true value of some parameter, β is 3. The power of a test is higher (i.e., we are more likely to reject the null hypothesis) if the value under the null β* is 1 rather than 2. Finally, it is sometimes possible to test a particular null hypothesis using several different approaches – for example, in Chapter 8 we will see that there are several different tests for unit roots based on different forms of the test statistic – and it is likely that different types of test will have different levels of power. An optimal test would be one with an actual test size that matched the nominal size and which had as high a power as possible. Such a test would imply, for example, that using a 5% significance level would result in the null being rejected exactly 5% of the time by chance alone, and that an incorrect null hypothesis would be rejected close to 100% of the time.

188

3.9 A Special Type of Hypothesis Test: The t-ratio Recall that the formula under a test of significance approach to hypothesis testing using a t-test for the slope parameter was (3.32) with the obvious adjustments to test a hypothesis about the intercept. If the test is

i.e., a test that the population parameter is zero against a two-sided alternative, this is known as a t-ratio test. Since β* = 0, the expression in equation (3.32) collapses to (3.33) Thus the ratio of the coefficient to its standard error, given by this expression, is known as the t-ratio or t-statistic. EXAMPLE 3.5 Suppose that we have calculated the estimates for the intercept and the slope (1.10 and –19.88, respectively) and their corresponding standard errors (1.35 and 1.98, respectively). The t-ratios associated with each of the intercept and slope coefficients would be given by

Note that if a coefficient is negative, its t-ratio will also be negative. In order to test (separately) the null hypotheses that α = 0 and β = 0, the test statistics would be compared with the appropriate critical value from a t-distribution. In this case, the number of degrees of freedom, given by T – k, is equal to 15 – 2 = 13. The 5% critical value for this two-sided test (remember, 2.5% in each tail for a 5% test) is 2.16, while 189

the 1% two-sided critical value (0.5% in each tail) is 3.01. Given these t-ratios and critical values, would the following null hypotheses be rejected?

If H0 is rejected, it would be said that the test statistic is significant. If the variable is not ‘significant’, it means that while the estimated value of the coefficient is not exactly zero (e.g. 1.10 in the example above), the coefficient is indistinguishable statistically from zero. If a zero were placed in the fitted equation instead of the estimated value, this would mean that whatever happened to the value of that explanatory variable, the dependent variable would be unaffected. This would then be taken to mean that the variable is not helping to explain variations in y, and that it could therefore be removed from the regression equation. For example, if the t-ratio associated with x had been –1.04 rather than – 10.04 (assuming that the standard error stayed the same), the variable would be classed as insignificant (i.e., not statistically different from zero). The only insignificant term in the above regression is the intercept. There are good statistical reasons for always retaining the constant, even if it is not significant; see Chapter 5. It is worth noting that, for degrees of freedom greater than around 25, the 5% two-sided critical value is approximately ±2. So, as a rule of thumb (i.e., a rough guide), the null hypothesis would be rejected if the t-statistic exceeds 2 in absolute value. Some authors place the t-ratios in parentheses below the corresponding coefficient estimates rather than the standard errors. One thus needs to check which convention is being used in each particular application, and also to state this clearly when presenting estimation results. There will now follow two finance case studies that involve only the estimation of bivariate linear regression models and the construction and interpretation of t-ratios.

3.10 An Example of a Simple t-test of a Theory in Finance: Can US Mutual Funds Beat the Market? Jensen (1968) was the first to systematically test the performance of 190

mutual funds, and in particular examine whether any ‘beat the market’. He used a sample of annual returns on the portfolios of 115 mutual funds from 1945 to 64. Each of the 115 funds was subjected to a separate OLS timeseries regression of the form (3.34) where Rjt is the return on portfolio j at time t, Rft is the return on a risk-free proxy (a one-year government bond), Rmt is the return on a market portfolio proxy, ujt is an error term, and αj, βj are parameters to be estimated. The quantity of interest is the significance of αj, since this parameter defines whether the fund outperforms or underperforms the market index. Thus the null hypothesis is given by: H0 : αj = 0. A positive and significant αj for a given fund would suggest that the fund is able to earn significant abnormal returns in excess of the market-required return for a fund of this given riskiness. This coefficient has become known as ‘Jensen’s alpha’. Some summary statistics across the 115 funds for the estimated regression results for equation (3.34) are given in Table 3.4. Table 3.4 Summary statistics for the estimated regression results for equation (3.34) Extremal values Item

Sample size

Mean value

Median value

Minimum

Maximum

–0.011

–0.009

–0.080

0.058

0.840

0.848

0.219

1.405

17

19

10

20

Source: Jensen (1968). Reprinted with the permission of Blackwell Publishers.

As Table 3.4 shows, the average (defined as either the mean or the median) fund was unable to ‘beat the market’, recording a negative alpha in both cases. There were, however, some funds that did manage to perform significantly better than expected given their level of risk, with the best fund of all yielding an alpha of 0.058. Interestingly, the average fund had a beta estimate of around 0.85, indicating that, in the CAPM context, 191

most funds were less risky than the market index. This result may be attributable to the funds investing predominantly in (mature) blue chip stocks rather than small caps. The most visual method of presenting the results was obtained by plotting the number of mutual funds in each t-ratio category for the alpha coefficient, first gross and then net of transactions costs, as in Figure 3.18 and Figure 3.19, respectively.

Figure 3.18 Frequency distribution of t-ratios of mutual fund alphas

(gross of transactions costs). Source: Jensen (1968). Reprinted with the permission of Blackwell Publishers

Figure 3.19 Frequency distribution of t-ratios of mutual fund alphas (net

of transactions costs). Source: Jensen (1968). Reprinted with the 192

permission of Blackwell Publishers The appropriate critical value for a two-sided test of αj = 0 is approximately 2.10 (assuming twenty years of annual data leading to eighteen degrees of freedom). As can be seen, only five funds have estimated t-ratios greater than 2 and are therefore implied to have been able to outperform the market before transactions costs are taken into account. Interestingly, five firms have also significantly underperformed the market, with t-ratios of −2 or less. When transactions costs are taken into account (Figure 3.19), only one fund out of 115 is able to significantly outperform the market, while 14 significantly underperform it. Given that a nominal 5% two-sided size of test is being used, one would expect two or three funds to ‘significantly beat the market’ by chance alone. It would thus be concluded that, during the sample period studied, US fund managers appeared unable to systematically generate positive abnormal returns.

3.11 Can UK Unit Trust Managers Beat the Market? Jensen’s study has proved pivotal in suggesting a method for conducting empirical tests of the performance of fund managers. However, it has been criticised on several grounds. One of the most important of these in the context of this book is that only between ten and twenty annual observations were used for each regression. Such a small number of observations is really insufficient for the asymptotic theory underlying the testing procedure to be validly invoked. A variant on Jensen’s test is now estimated in the context of the UK market, by considering monthly returns on seventy-six equity unit trusts. The data cover the period January 1979–May 2000 (257 observations for each fund). Some summary statistics for the funds are presented in Table 3.5. Table 3.5 Summary statistics for unit trust returns, January 1979–May 2000

Average monthly

Mean (%)

Minimum (%)

Maximum (%)

Median (%)

1.0

0.6

1.4

1.0

193

return, 1979–2000 Standard deviation of returns over time

5.1

4.3

6.9

5.0

From these summary statistics, the average continuously compounded return is 1% per month, although the most interesting feature is the wide variation in the performances of the funds. The worst-performing fund yields an average return of 0.6% per month over the twenty-year period, while the best would give 1.4% per month. This variability is further demonstrated in Figure 3.20, which plots over time the value of £100 invested in each of the funds in January 1979.

Figure 3.20 Performance of UK unit trusts, 1979–2000

A regression of the form (3.34) is applied to the UK data, and the summary results presented in Table 3.6. A number of features of the regression results are worthy of further comment. First, most of the funds have estimated betas less than one again, perhaps suggesting that the fund managers have historically been risk averse or investing disproportionately in blue chip companies in mature sectors. Second, gross of transactions costs, nine funds of the sample of seventy-six were able to significantly 194

outperform the market by providing a significant positive alpha, while seven funds yielded significant negative alphas. The average fund (where ‘average’ is measured using either the mean or the median) is not able to earn any excess return over the required rate given its level of risk. Table 3.6 CAPM regression results for unit trust returns, January 1979– May 2000

3.12 The Overreaction Hypothesis and the UK Stock Market 3.12.1 Motivation Two studies by DeBondt and Thaler (1985, 1987) showed that stocks experiencing a poor performance over a three to five-year period subsequently tend to outperform stocks that had previously performed relatively well. This implies that, on average, stocks which are ‘losers’ in terms of their returns subsequently become ‘winners’, and vice versa. This chapter now examines a paper by Clare and Thomas (1995) that conducts a similar study using monthly UK stock returns from January 1955 to 1990 (thirty-six years) on all firms traded on the London Stock Exchange (LSE). This phenomenon seems at first blush to be inconsistent with the efficient markets hypothesis, and Clare and Thomas (1995) propose two explanations (see Box 3.9). Zarowin (1990) also finds that 80% of the extra return available from holding the losers accrues to investors in January, so that almost all of the ‘overreaction effect’ seems to occur at the start of the calendar year. BOX 3.9 Reasons for stock market overreactions (1) That the ‘overreaction effect’ is just another manifestation of the ‘size effect’. The size effect is the tendency of small firms to generate, on average, superior returns to large firms. The argument would follow that the losers were small firms and that these small firms would subsequently outperform the large 195

firms. DeBondt and Thaler did not believe this to be a sufficient explanation, but Zarowin (1990) found that allowing for firm size did reduce the subsequent return on the losers. (2) That the reversals of fortune reflect changes in equilibrium required returns. The losers are argued to be likely to have considerably higher CAPM betas, reflecting investors’ perceptions that they are more risky. Of course, betas can change over time, and a substantial fall in the firms’ share prices (for the losers) would lead to a rise in their leverage ratios, leading in all likelihood to an increase in their perceived riskiness. Therefore, the required rate of return on the losers will be larger, and their ex post performance better. Ball and Kothari (1989) find the CAPM betas of losers to be considerably higher than those of winners.

3.12.2 Methodology Clare and Thomas (1995) take a random sample of 1000 firms and, for each, they calculate the monthly excess return of the stock for the market over a twelve-, twentyfour- or thirty-six-month period for each stock i (3.35) Then the average monthly return over each stock i for the first twelve-, twenty-four-, or thirty-six-month period is calculated (3.36) The stocks are then ranked from highest average return to lowest and from these five portfolios are formed and returns are calculated assuming an equal weighting of stocks in each portfolio (Box 3.10). BOX 3.10 Ranking stocks and forming portfolios Portfolio Portfolio 1 Portfolio 2

Ranking Best performing 20% of firms Next 20% 196

Portfolio 3 Portfolio 4 Portfolio 5

Next 20% Next 20% Worst performing 20% of firms

The same sample length n is used to monitor the performance of each portfolio. Thus, for example, if the portfolio formation period is one, two or three years, the subsequent portfolio tracking period will also be one, two or three years, respectively. Then another portfolio formation period follows and so on until the sample period has been exhausted. How many samples of length n will there be? n = 1, 2 or 3 years. First, suppose n = 1 year. The procedure adopted would be as shown in Box 3.11. BOX 3.11 Portfolio monitoring Estimate for year 1 Monitor portfolios for year 2 Estimate for year 3 ⋮ Monitor portfolios for year 36 So if n = 1, there are eighteen independent (non-overlapping) observation periods and eighteen independent tracking periods. By similar arguments, n = 2 gives nine independent periods and n = 3 gives six independent periods. The mean return for each month over the 18, 9 or 6 periods for the winner and loser portfolios (the top 20% and bottom 20% of firms in the portfolio formation period) are denoted by and respectively. Define the difference between these as The first regression to be performed is of the excess return of the losers over the winners on a constant only (3.37) where ηt is an error term. The test is of whether α1 is significant and positive. However, a significant and positive α1 is not a sufficient condition for the over-reaction effect to be confirmed, because it could be owing to higher returns being required on loser stocks owing to loser stocks being more risky. The solution, Clare and Thomas, (1995) argue, is 197

to allow for risk differences by regressing against the market risk premium (3.38) where Rmt is the return on the FTA All-Share, and Rft is the return on a UK government three-month Treasury Bill. The results for each of these two regressions are presented in Table 3.7. Table 3.7 Is there an overreaction effect in the UK stock market? Panel A: all months n = 12

n = 24

n = 36

Return on loser

0.0033

0.0011

0.0129

Return on winner

0.0036

–0.0003

0.0115

Implied annualised return difference

–0.37%

1.68%

1.56%

Coefficient for (3.37):

–0.00031

0.0014**

0.0013*

(–0.29)

(2.01)

(1.55)

–0.00034

0.00147**

0.0013

(–0.30)

(2.01)

(1.41)

–0.022

0.010

–0.0025

(–0.25)

(0.21)

(–0.06)

–0.0007

0.0012*

0.0009

(–0.72)

(1.63)

(1.05)

Coefficients for (3.38): Coefficients for (3.38):

Panel B: all months except January Coefficient for (3.37):

Notes: t-ratios in parentheses;* and** denote significance at the 10% and 5% levels, respectively. Source: Clare and Thomas (1995). Reprinted with the permission of Blackwell Publishers.

198

As can be seen by comparing the returns on the winners and losers in the first two rows of Table 3.7, twelve months is not a sufficiently long time for losers to become winners. By the two-year tracking horizon, however, the losers have become winners, and similarly for the three-year samples. This translates into an average 1.68% higher return on the losers than the winners at the two-year horizon, and 1.56% higher return at the three-year horizon. Recall that the estimated value of the coefficient in a regression of a variable on a constant only is equal to the average value of that variable. It can also be seen that the estimated coefficients on the constant terms for each horizon are exactly equal to the differences between the returns of the losers and the winners. This coefficient is statistically significant at the two-year horizon, and marginally significant at the three-year horizon. In the second test regression, represents the difference between the market betas of the winner and loser portfolios. None of the beta coefficient estimates are even close to being significant, and the inclusion of the risk term makes virtually no difference to the coefficient values or significances of the intercept terms. Removal of the January returns from the samples reduces the subsequent degree of overperformance of the loser portfolios, and the significances of the terms is somewhat reduced. It is concluded, therefore, that only a part of the overreaction phenomenon occurs in January. Clare and Thomas (1995), then proceed to examine whether the overreaction effect is related to firm size, although the results are not presented here.

3.12.3 Conclusions The main conclusions from Clare and Thomas’ study are: (1) There appears to be evidence of overreactions in UK stock returns, as found in previous US studies (2) These overreactions are unrelated to the CAPM beta (3) Losers that subsequently become winners tend to be small, so that most of the overreaction in the UK can be attributed to the size effect

3.13 The Exact Significance Level The exact significance level is also commonly known as the p-value. It gives the marginal significance level where one would be indifferent 199

between rejecting and not rejecting the null hypothesis. If the test statistic is ‘large’ in absolute value, the p-value will be small, and vice versa. For example, consider a test statistic that is distributed as a t62 and takes a value of 1.47. Would the null hypothesis be rejected? It would depend on the size of the test. Now, suppose that the p-value for this test is calculated to be 0.12 Is the null rejected at the 5% level? No Is the null rejected at the 10% level? No Is the null rejected at the 20% level? Yes In fact, the null would have been rejected at the 12% level or higher. To see this, consider conducting a series of tests with size 0.1%, 0.2%, 0.3%, 0.4%, …1%, …, 5%, …10%, …Eventually, the critical value and test statistic will meet and this will be the p-value. p-values are almost always provided automatically by software packages. Note how useful they are! They provide all of the information required to conduct a hypothesis test without requiring of the researcher the need to calculate a test statistic or to find a critical value from a table – both of these steps have already been taken by the package in producing the p-value. The p-value is also useful since it avoids the requirement of specifying an arbitrary significance level (α). Sensitivity analysis of the effect of the significance level on the conclusion occurs automatically. Informally, the p-value is also often referred to as the probability of being wrong when the null hypothesis is rejected. Thus, for example, if a p-value of 0.05 or less leads the researcher to reject the null (equivalent to a 5% significance level), this is equivalent to saying that if the probability of incorrectly rejecting the null is more than 5%, do not reject it. The pvalue has also been termed the ‘plausibility’ of the null hypothesis; so, the smaller is the p-value, the less plausible is the null hypothesis. KEY CONCEPTS The key terms to be able to define and explain from this chapter are regression model population linear model unbiasedness standard error 200

null hypothesis t-distribution test statistic type I error size of a test p-value disturbance term sample consistency efficiency statistical inference alternative hypothesis confidence interval rejection region type II error power of a test asymptotic

Appendix 3.1 Mathematical Derivations of CLRM Results 3A.1 Derivation of the OLS Coefficient Estimator in the Bivariate Case (3A.1) It is necessary to minimise L w.r.t. and to find the values of α and β that give the line that is closest to the data. So L is differentiated w.r.t. and and the first derivatives are set to zero. The first derivatives are given by (3A.2) (3A.3) The next step is to rearrange equations (3A.2) and (3A.3) in order to obtain expressions for and From equation (3A.2) 201

(3A.4) Expanding the parentheses and recalling that the sum runs from 1 to T so that there will be T terms in (3A.5) But

and

so it is possible to write equation (3A.5) as (3A.6)

or (3A.7) From equation (3A.3) (3A.8) equation From (3A.7) (3A.9) Substituting into equation (3A.8) for

from equation (3A.9) (3A.10) (3A.11) (3A.12)

Rearranging for (3A.13) Dividing both sides of equation (3A.13) by

gives (3A.14)

202

3A.2 Derivation of the OLS Standard Error Estimators for the Intercept and Slope in the Bivariate Case Recall that the variance of the random variable

can be written as (3A.15)

and since the OLS estimator is unbiased (3A.16) By similar arguments, the variance of the slope estimator can be written as (3A.17) Working first with equation (3A.17), replacing given by the OLS estimator

with the formula for it

(3A.18) Replacing yt with α + βxt + ut, and replacing (3A.18)

with

in equation

(3A.19) Cancelling α and multiplying the last β term in equation (3A.19) by

(3A.20) Rearranging (3A.21)

203

(3A.22)

Now the β terms equation in (3A.22) will cancel to give (3A.23) Now let denote the mean-adjusted observation for xt, i.e. Equation (3A.23) can be written (3A.24) The denominator of equation (3A.24) can be taken through the expectations operator under the assumption that x is fixed or nonstochastic (3A.25) Writing the terms out in the last summation of equation (3A.25) (3A.26) Now expanding the brackets of the squared term in the expectations operator of equation (3A.26) (3A.27) where ‘cross-products’ in equation (3A.27) denotes all of the terms These cross-products can be written as and their expectation will be zero under the assumption that the error terms are uncorrelated with one another. Thus, the ‘cross-products’ term in equation (3A.27) will drop out. Recall also from the chapter text that is the error variance, which is estimated using s2 204

(3A.28) which can also be written (3A.29) A term in can be cancelled from the numerator and denominator of equation (3A.29), and recalling that this gives the variance of the slope coefficient as (3A.30) so that the standard error can be obtained by taking the square root of (3A.30) (3A.31) Turning now to the derivation of the intercept standard error, this is much more difficult than that of the slope standard error. In fact, both are very much easier using matrix algebra as shown below. Therefore, this derivation will be offered in summary form. It is possible to express as a function of the true α and of the disturbances, ut

(3A.32)

Denoting all of the elements in square brackets as gt, equation (3A.32) can be written (3A.33) From equation (3A.15), the intercept variance would be written 205

(3A.34) Writing equation (3A.34) out in full for

and expanding the brackets

(3A.35)

This looks rather complex, but fortunately, if we take outside the square brackets in the numerator, the remaining numerator cancels with a term in the denominator to leave the required result (3A.36)

SELF-STUDY QUESTIONS 1. (a) Why does OLS estimation involve taking vertical deviations of the points to the line rather than horizontal distances? (b) Why are the vertical distances squared before being added together? (c) Why are the squares of the vertical distances taken rather than the absolute values? 2. Explain, with the use of equations, the difference between the sample regression function and the population regression function. 3. What is an estimator? Is the OLS estimator superior to all other estimators? Why or why not? 4. What five assumptions are usually made about the unobservable error terms in the classical linear regression model (CLRM)? Briefly explain the meaning of each. Why are these assumptions made? 5. Which of the following models can be estimated (following a suitable rearrangement if necessary) using ordinary least squares (OLS), where X, y, Z are variables and α, β, γ are parameters to be estimated? (Hint: the models need to be linear in the parameters.) 206

(3.39) (3.40) (3.41) (3.42) (3.43) 6. The capital asset pricing model (CAPM) can be written as (3.44) using the standard notation. The first step in using the CAPM is to estimate the stock’s beta using the market model. The market model can be written as (3.45) where Rit is the excess return for security i at time t, Rmt is the excess return on a proxy for the market portfolio at time t, and ut is an iid random disturbance term. The cofficient beta in this case is also the CAPM beta for security i. Suppose that you had estimated equation (3.45) and found that the estimated value of beta for a stock, was 1.147. The standard error associated with this coefficient is estimated to be 0.0548. A city analyst has told you that this security closely follows the market, but that it is no more risky, on average, than the market. This can be tested by the null hypotheses that the value of beta is one. The model is estimated over sixty-two daily observations. Test this hypothesis against a one-sided alternative that the security is more risky than the market, at the 5% level. Write down the null and alternative hypothesis. What do you conclude? Are the analyst’s claims empirically verified? 7. The analyst also tells you that shares in Chris Mining plc have no systematic risk, in other words that the returns on its shares are 207

completely unrelated to movements in the market. The value of beta and its standard error are calculated to be 0.214 and 0.186, respectively. The model is estimated over thirty-eight quarterly observations. Write down the null and alternative hypotheses. Test this null hypothesis against a two-sided alternative. 8. Form and interpret a 95% and a 99% confidence interval for beta using the figures given in Question 7. 9. Are hypotheses tested concerning the actual values of the coefficients (i.e., β) or their estimated values (i.e., ) and why?

1 2

Strictly, the assumption that the xs are non-stochastic is stronger than required, an issue that will be discussed in more detail in Chapter 5. Strictly, these are the estimated standard errors conditional on the parameter estimates, and so should be denoted and but the additional layer of hats will be omitted here since the meaning should be obvious from the context.

208

4 Further Development and Analysis of the Classical Linear Regression Model

LEARNING OUTCOMES In this chapter, you will learn how to Construct models with more than one explanatory variable Test multiple hypotheses using an F-test Determine how well a model fits the data Form a restricted regression Derive the OLS parameter and standard error estimators using matrix algebra Construct and interpret quantile regression models

4.1 Generalising the Simple Model to Multiple Linear Regression Previously, a model of the following form has been used (4.1) Equation (4.1) is a simple bivariate regression model. That is, changes in the dependent variable are explained by reference to changes in one single explanatory variable x. But what if the financial theory or idea that is sought to be tested suggests that the dependent variable is influenced by more than one independent variable? For example, simple estimation and tests of the capital asset pricing model (CAPM) can be conducted using an 209

equation of the form of (4.1), but arbitrage pricing theory does not presuppose that there is only a single factor affecting stock returns. So, to give one illustration, stock returns might be purported to depend on their sensitivity to unexpected changes in: (1) (2) (3) (4)

inflation differences in returns on short- and long-dated bonds industrial production default risks

Having just one independent variable would be no good in this case. It would of course be possible to use each of the four proposed explanatory factors in separate regressions. But it is of greater interest and it is more valid to have more than one explanatory variable in the regression equation at the same time, and therefore to examine the effect of all of the explanatory variables together on the explained variable. It is very easy to generalise the simple model to one with k regressors (independent variables). Equation (4.1) becomes (4.2) So the variables x2t, x3t, …, xkt are a set of k – 1 explanatory variables which are thought to influence y, and the coefficient estimates β1, β2, …, βk are the parameters which quantify the effect of each of these explanatory variables on y. The coefficient interpretations are slightly altered in the multiple regression context. Each coefficient is now known as a partial regression coefficient, interpreted as representing the partial effect of the given explanatory variable on the explained variable, after holding constant, or eliminating the effect of, all other explanatory variables. For example, measures the effect of x2 on y after eliminating the effects of x3, x4, …, xk. Stating this in other words, each coefficient measures the average change in the dependent variable per unit change in a given independent variable, holding all other independent variables constant at their average values.

4.2 The Constant Term In equation (4.2) above, astute readers will have noticed that the explanatory variables are numbered x2, x3, … i.e., the list starts with x2 and 210

not x1. So, where is x1? In fact, it is the constant term, usually represented by a column of ones of length T:

(4.3)

Thus there is a variable implicitly hiding next to β1, which is a column vector of ones, the length of which is the number of observations in the sample. The x1 in the regression equation is not usually written, in the same way that one unit of p and two units of q would be written as ‘p + 2q’ and not ‘1p + 2q’. β1 is the coefficient attached to the constant term (which was called α in Chapter 3). This coefficient can still be referred to as the intercept, which can be interpreted as the average value which y would take if all of the explanatory variables took a value of zero. A tighter definition of k, the number of explanatory variables, is probably now necessary. Throughout this book, k is defined as the number of ‘explanatory variables’ or ‘regressors’ including the constant term. This is equivalent to the number of parameters that are estimated in the regression equation. Strictly speaking, it is not sensible to call the constant an explanatory variable, since it does not explain anything and it always takes the same values. However, this definition of k will be employed for notational convenience. Equation (4.2) can be expressed even more compactly by writing it in matrix form (4.4) where: y X β u

is of dimension T × 1 is of dimension T × k is of dimension k × 1 is of dimension T × 1

The difference between (4.2) and (4.4) is that all of the time observations have been stacked up in a vector, and also that all of the different explanatory variables have been squashed together so that there is a column for each in the X matrix. Such a notation may seem unnecessarily 211

complex, but in fact, the matrix notation is usually more compact and convenient. So, for example, if k is 2, i.e., there are two regressors, one of which is the constant term (equivalent to a simple bivariate regression yt = α + βxt + ut), it is possible to write

(4.5)

so that the xij element of the matrix X represents the jth time observation on the ith variable. Notice that the matrices written in this way are conformable – in other words, there is a valid matrix multiplication and addition on the RHS. The above presentation is the standard way to express matrices in the time-series econometrics literature, although the ordering of the indices is different to that used in the mathematics of matrix algebra (as presented in Chapter 1 of this book). In the latter case, xij would represent the element in row i and column j, although in the notation used in the body of this book it is the other way around.

4.3 How are the Parameters (the Elements of the β Vector) Calculated in the Generalised Case? Previously, the residual sum of squares, was minimised with respect to α and β. In the multiple regression context, in order to obtain estimates of the parameters, β1, β2, …, βk, the RSS would be minimised with respect to all the elements of β. Now, the residuals can be stacked in a vector: (4.6)

The RSS is still the relevant loss function, and would be given in a matrix notation by

212

(4.7)

Using a similar procedure to that employed in the bivariate regression case, i.e., substituting into equation (4.7), and denoting the vector of estimated parameters as it can be shown (see Appendix 4.1 to this chapter) that the coefficient estimates will be given by the elements of the expression

(4.8)

If one were to check the dimensions of the RHS of equation (4.8), it would be observed to be k × 1. This is as required since there are k parameters to be estimated by the formula for But how are the standard errors of the coefficient estimates calculated? Previously, to estimate the variance of the errors, σ2, an estimator denoted by s2 was used (4.9) The denominator of equation (4.9) is given by T – 2, which is the number of degrees of freedom for the bivariate regression model (i.e., the number of observations minus two). This essentially applies since two observations are effectively ‘lost’ in estimating the two model parameters (i.e., in deriving estimates for α and β). In the case where there is more than one explanatory variable plus a constant, and using the matrix notation, equation (4.9) would be modified to (4.10) where k = number of regressors including a constant. In this case, k observations are ‘lost’ as k parameters are estimated, leaving T – k degrees of freedom. It can also be shown (see Appendix 4.1 to this chapter) that the 213

parameter variance–covariance matrix is given by (4.11) The leading diagonal terms give the coefficient variances while the offdiagonal terms give the covariances between the parameter estimates, so that the variance of is the first diagonal element, the variance of is the second element on the leading diagonal, and the variance of is the kth diagonal element. The coefficient standard errors are thus simply given by taking the square roots of each of the terms on the leading diagonal.

4.4 Testing Multiple Hypotheses: The F-test The t-test was used to test single hypotheses, i.e., hypotheses involving only one coefficient. But what if it is of interest to test more than one coefficient simultaneously? For example, what if a researcher wanted to determine whether a restriction that the coefficient values for β2 and β3 are both unity could be imposed, so that an increase in either one of the two variables x2 or x3 would cause y to rise by one unit? The t-testing framework is not sufficiently general to cope with this sort of hypothesis test. Instead, a more general framework is employed, centring on an F-test. Under the F-test framework, two regressions are required, known as the unrestricted and the restricted regressions. The unrestricted regression is the one in which the coefficients are freely determined by the data, as has been constructed previously. The restricted regression is the one in which the coefficients are restricted, i.e., the restrictions are imposed on some βs. Thus the F-test approach to hypothesis testing is also termed restricted least squares, for obvious reasons. The residual sums of squares from each regression are determined, and the two residual sums of squares are ‘compared’ in the test statistic. The Ftest statistic for testing multiple hypotheses about the coefficient estimates is given by (4.12) where the following notation applies: EXAMPLE 4.1

214

The following model with three regressors (including the constant, so k = 3) is estimated over fifteen observations (4.13) and the following data have been calculated from the original xs

Calculate the coefficient estimates and their standard errors.

(4.14)

To calculate the standard errors, an estimate of σ2 is required (4.15) The variance–covariance matrix of

is given by (4.16)

The coefficient variances are on the diagonals, and the standard errors are found by taking the square roots of each of the coefficient variances (4.17) (4.18) (4.19)

215

The estimated equation would be written (4.20)

Fortunately, in practice all econometrics software packages will estimate the coefficient values and their standard errors. Clearly, though, it is still useful to understand where these estimates came from.

The most important part of the test statistic to understand is the numerator expression RRSS – URSS. To see why the test centres around a comparison of the residual sums of squares from the restricted and unrestricted regressions, recall that OLS estimation involved choosing the model that minimised the residual sum of squares, with no constraints imposed. Now if, after imposing constraints on the model, a residual sum of squares results that is not much higher than the unconstrained model’s residual sum of squares, it would be concluded that the restrictions were supported by the data. On the other hand, if the residual sum of squares increased considerably after the restrictions were imposed, it would be concluded that the restrictions were not supported by the data and therefore that the hypothesis should be rejected. It can be further stated that RRSS ≥ URSS. Only under a particular set of very extreme circumstances will the residual sums of squares for the restricted and unrestricted models be exactly equal. This would be the case when the restriction was already present in the data, so that it is not really a restriction at all (it would be said that the restriction is ‘not binding’, i.e., it does not make any difference to the parameter estimates). So, for example, if the null hypothesis is H0: β2 = 1 and β3 = 1, then RRSS = URSS only in the case where the coefficient estimates for the unrestricted regression had been and Of course, such an event is extremely unlikely to occur in practice. 216

It is worth noting that the F-test statistic is sometimes expressed in a slightly different way, which is simply a rearrangement of (4.12) as

Sometimes, m is termed the degrees of freedom for the numerator while T – k are the degrees of freedom for the denominator and writing the formula for the F-test statistic in this way makes it easy to see why. EXAMPLE 4.2 Dropping the time subscripts for simplicity, suppose that the general regression is (4.21) and that the restriction β3 + β4 = 1 is under test (there exists some hypothesis from theory which suggests that this would be an interesting hypothesis to study). The unrestricted regression is equation (4.21) above, but what is the restricted regression? It could be expressed as (4.22) The restriction (β3 + β4 = 1) is substituted into the regression so that it is automatically imposed on the data. The way that this would be achieved would be to make either β3 or β4 the subject of equation (4.22), e.g. (4.23) and then substitute into equation (4.21) for β4 (4.24) Equation (4.24) is already a restricted form of the regression, but it is not yet in the form that is required to estimate it using a computer package. In order to be able to estimate a model using OLS, software packages usually require each RHS variable to be multiplied by one coefficient only. Therefore, a little more algebraic manipulation is required. First, expanding the brackets around (1 – β3) 217

(4.25) Then, gathering all of the terms in each βi together and rearranging (4.26) Note that any variables without coefficients attached (e.g. x4 in equation (4.25)) are taken over to the LHS and are then combined with y. Equation (4.26) is the restricted regression. It is actually estimated by creating two new variables – call them, say, P and Q, where P = y – x4 and Q = x3 – x4 – so the regression that is actually estimated is (4.27) What would have happened if instead β3 had been made the subject of equation (4.23) and β3 had therefore been removed from the equation? Although the equation that would have been estimated would have been different from equation (4.27), the value of the residual sum of squares for these two models (both of which have imposed upon them the same restriction) would be the same. The test statistic follows the F-distribution under the null hypothesis. The F-distribution has two degrees of freedom parameters (recall that the t-distribution had only one degree of freedom parameter, equal to T – k). The value of the degrees of freedom parameters for the F-test are m, the number of restrictions imposed on the model, and (T – k), the number of observations less the number of regressors for the unrestricted regression, respectively. Note that the order of the degree of freedom parameters is important. The appropriate critical value will be in column m, row (T – k) of the F-distribution tables.

4.4.1 The Relationship Between the t- and the F-Distributions Any hypothesis that could be tested with a t-test could also have been tested using an F-test, but not the other way around. So, single hypotheses involving one coefficient can be tested using a t- or an F-test, but multiple hypotheses can be tested only using an F-test. For example, consider the hypothesis

218

This hypothesis could have been tested using the usual t-test (4.28) or it could be tested in the framework above for the F-test. Note that the two tests always give the same conclusion since the t-distribution is just a special case of the F-distribution. For example, consider any random variable Z that follows a t-distribution with T – k degrees of freedom, and square it. The square of the t is equivalent to a particular form of the Fdistribution

Thus the square of a t-distributed random variable with T – k degrees of freedom also follows an F-distribution with 1 and T – k degrees of freedom. This relationship between the t and the F-distributions will always hold – take some examples from the statistical tables and try it! The F-distribution has only positive values and is not symmetrical. Therefore, the null is rejected only if the test statistic exceeds the critical F-value, although the test is a two-sided one in the sense that rejection will occur if is significantly bigger or significantly smaller than 0.5.

4.4.2 Determining the Number of Restrictions, m How is the appropriate value of m decided in each case? Informally, the number of restrictions can be seen as ‘the number of equality signs under the null hypothesis’. To give some examples

At first glance, you may have thought that in the first of these cases, the number of restrictions was two. In fact, there is only one restriction that involves two coefficients. The number of restrictions in the second two examples is obvious, as they involve two and three separate component restrictions, respectively. The last of these three examples is particularly important. If the model is 219

(4.29) then the null hypothesis of is tested by ‘THE’ regression F-statistic. It tests the null hypothesis that all of the coefficients except the intercept coefficient are zero. This test is sometimes called a test for ‘junk regressions’, since if this null hypothesis cannot be rejected, it would imply that none of the independent variables in the model was able to explain variations in y. Note the form of the alternative hypothesis for all tests when more than one restriction is involved In other words, ‘and’ occurs under the null hypothesis and ‘or’ under the alternative, so that it takes only one part of a joint null hypothesis to be wrong for the null hypothesis as a whole to be rejected.

4.4.3 Hypotheses that Cannot be Tested with Either an F- or a t-Test It is not possible to test hypotheses that are not linear or that are multiplicative using this framework – for example, H0 : β2β3 = 2, or H0 : cannot be tested. EXAMPLE 4.3 Suppose that a researcher wants to test whether the returns on a company stock (y) show unit sensitivity to two factors (factor x2 and factor x3) among three considered. The regression is carried out on 144 monthly observations. The regression is (4.30) (1) What are the restricted and unrestricted regressions? (2) If the two RSS are 436.1 and 397.2, respectively, perform the test. Unit sensitivity to factors x2 and x3 implies the restriction that the coefficients on these two variables should be unity, so H0: β2 = 1 and 220

β3 = 1. The unrestricted regression will be the one given by equation (4.30) above. To derive the restricted regression, first impose the restriction: (4.31) Replacing β2 and β3 by their values under the null hypothesis (4.32) Rearranging (4.33) Defining z = y – x2 – x3, the restricted regression is one of z on a constant and x4 (4.34) The formula for the F-test statistic is given in equation (4.12) above. For this application, the following inputs to the formula are available: T = 144, k = 4, m = 2, RRSS = 436.1, URSS = 397.2. Plugging these into the formula gives an F-test statistic value of 6.86. This statistic should be compared with an F(m, T – k), which in this case is an F(2, 140). The critical values are 3.07 at the 5% level and 4.79 at the 1% level. Note that the table does not include a row for 140, so we use the closest, which is 120 rather than ∞. The test statistic clearly exceeds the critical values at both the 5% and 1% levels, and hence the null hypothesis is rejected. It would thus be concluded that the restriction is not supported by the data.

4.4.4 A Note on Sample Sizes and Asymptotic Theory A question that is often asked by those new to econometrics is ‘what is an appropriate sample size for model estimation?’ While there is no definitive answer to this question, it should be noted that most testing procedures in econometrics rely on asymptotic theory. That is, the results in theory hold only if there is an infinite number of observations. In practice, an infinite number of observations will never be available and fortunately, an infinite 221

number of observations are not usually required to invoke the asymptotic theory. An approximation to the asymptotic behaviour of the test statistics can be obtained using finite samples, provided that they are large enough. In general, as many observations as possible should be used (although there are important caveats to this statement relating to ‘structural stability’, discussed in Chapter 5). The reason is that all the researcher has at his disposal is a sample of data from which to estimate parameter values and to infer their likely population counterparts. A sample may fail to deliver something close to the exact population values owing to sampling error. Even if the sample is randomly drawn from the population, some samples will be more representative of the behaviour of the population than others, purely owing to ‘luck of the draw’. Sampling error is minimised by increasing the size of the sample, since the larger the sample, the less likely it is that all of the data drawn will be unrepresentative of the population.

4.5 Data Mining and the True Size of the Test Recall that the probability of rejecting a correct null hypothesis is equal to the size of the test, denoted α. The possibility of rejecting a correct null hypothesis arises from the fact that test statistics are assumed to follow a random distribution and hence they will take on extreme values that fall in the rejection region some of the time by chance alone. A consequence of this is that it will almost always be possible to find significant relationships between variables if enough variables are examined. For example, suppose that a dependent variable yt and twenty explanatory variables x2t, …, x21t (excluding a constant term) are generated separately as independent normally distributed random variables. Then y is regressed separately on each of the twenty explanatory variables plus a constant, and the significance of each explanatory variable in the regressions is examined. If this experiment is repeated many times, on average one of the twenty regressions will have a slope coefficient that is significant at the 5% level for each experiment. The implication is that for any regression, if enough explanatory variables are employed in a regression, often one or more will be significant by chance alone. More concretely, it could be stated that if an α% size of test is used, on average one in every (100/α) regressions will have a significant slope coefficient by chance alone. Trying many variables in a regression without basing the selection of the candidate variables on a financial or economic theory is known as ‘data mining’ or ‘data snooping’. The result in such cases is that the true 222

significance level will be considerably greater than the nominal significance level assumed. For example, suppose that twenty separate regressions are conducted, of which three contain a significant regressor, and a 5% nominal significance level is assumed, then the true significance level would be much higher (e.g., 25%). Therefore, if the researcher then shows only the results for the regression containing the final three equations and states that they are significant at the 5% level, inappropriate conclusions concerning the significance of the variables would result. As well as ensuring that the selection of candidate regressors for inclusion in a model is made on the basis of financial or economic theory, another way to avoid data mining is by examining the forecast performance of the model in an ‘out-of-sample’ data set (see Chapter 6). The idea is essentially that a proportion of the data is not used in model estimation, but is retained for model testing. A relationship observed in the estimation period that is purely the result of data mining, and is therefore spurious, is very unlikely to be repeated for the out-of-sample period. Therefore, models that are the product of data mining are likely to fit very poorly and to give very inaccurate forecasts for the out-of-sample period.

4.6 Qualitative Variables There are many situations when building an econometric model where we would like to capture the effect of qualitative information. For example, we might be interested in modelling credit ratings, or comparing the performance of men versus women traders to determine who takes more risk on average. In both these situations (credit ratings and sex of traders), the data do not have numbers associated with them initially. The way that we would turn qualitative information into quantitative variables that can be incorporated into the model is via the construction of one or more dummy variables. Dummy variables are usually specified to take on one of a narrow range of integer values, and in most instances only zero and one are used. Dummy variables can be used in the context of cross-sectional or timeseries regressions. In each case, the dummy variables are used in the same way as other explanatory variables and the coefficients on the dummy variables can be interpreted as the average differences in the values of the dependent variable for each category, given all of the other factors in the model. For example, suppose we have data and estimate the following regression model 223

(4.35) where salaryi is the annual salary in US dollars of trader i, age is his or her age, sex is his or her sex (with 1 = male; 0 = female), location = 0 if the trader is based in New York, 1 if he or she is based in London, 2 if he or she is based in Paris, and edu = 1 if the trader has a first degree or higher and 0 otherwise. In this case, all four of the explanatory variables in the model would be dummies and three of them take only values 0 and 1. For the latter, the coefficients on the dummies would be easy to interpret as the average difference in salary between a trader with this characteristic and an otherwise identical one without. For example, suppose that this would suggest that on average male traders (dummy value 1) earn $2850 per year more than otherwise equivalent female traders. Similarly, if the estimated value of β5 is this would show that traders having at least a degree earn on average $8500 per year more than those without. But what about the location dummy? This is more tricky to interpret and in fact it would probably be inappropriate to set up a dummy in this way. The dummy has three values, and thus implies an ordinal ranking of numbers which was probably not intended. Thus instead we would need to set up two or three separate 0–1 dummy variables, assuming that all traders in our sample are based in one of these locations and no other places. So, for example, we could have a NYi dummy variable that took the value 1 if the trader is based in New York and 0 otherwise; similarly we would set up a London dummy and a Paris dummy. We could either include all three dummy variables together (and not include an intercept in the regression equation) or only include two of the dummy variables and retain the intercept. If we include all three dummy variables and the intercept at the same time, the regression model could not be estimated and this is known as the dummy variable trap – see Section 10.3 in Chapter 10 for details. All of the above variables are known as intercept dummies since, effectively, they modify the intercept in each case (e.g., allowing for a different intercept or average salary for men versus women) but they do not alter the relationship between the dependent variable and the other independent variables – the latter would be known as slope dummy variables, which are discussed in Section 10.3 of Chapter 10. It should be evident given this brief introduction that dummy variables are extremely useful and will be used extensively in other parts of the 224

book, but most notably to allow for outliers in Chapter 5, to account for seasonality in Chapter 10 and when investigating limited dependent variables, which is the subject of the whole of Chapter 12.

4.7 Goodness of Fit Statistics 4.7.1 R2 It is desirable to have some measure of how well the regression model actually fits the data. In other words, it is desirable to have an answer to the question, ‘how well does the model containing the explanatory variables that was proposed actually explain variations in the dependent variable?’ Quantities known as goodness of fit statistics are available to test how well the sample regression function (SRF) fits the data – that is, how ‘close’ the fitted regression line is to all of the data points taken together. Note that it is not possible to say how well the sample regression function fits the population regression function – i.e., how the estimated model compares with the true relationship between the variables, since the latter is never known. But what measures might make plausible candidates to be goodness of fit statistics? A first response to this might be to look at the residual sum of squares (RSS). Recall that OLS selected the coefficient estimates that minimised this quantity, so the lower was the minimised value of the RSS, the better the model fitted the data. Consideration of the RSS is certainly one possibility, but RSS is unbounded from above (strictly, RSS is bounded from above by the total sum of squares – see below) – i.e., it can take any (non-negative) value. So, for example, if the value of the RSS under OLS estimation was 136.4, what does this actually mean? It would therefore be very difficult, by looking at this number alone, to tell whether the regression line fitted the data closely or not. The value of RSS depends to a great extent on the scale of the dependent variable. Thus, one way to pointlessly reduce the RSS would be to divide all of the observations on y by 10! In fact, a scaled version of the residual sum of squares is usually employed. The most common goodness of fit statistic is known as R2. One way to define R2 is to say that it is the square of the correlation coefficient between y and – that is, the square of the correlation between the values of the dependent variable and the corresponding fitted values from the model. A correlation coefficient must lie between –1 and +1 by definition. Since R2 defined in this way is the square of a correlation coefficient, it 225

must lie between 0 and 1. If this correlation is high, the model fits the data well, while if the correlation is low (close to zero), the model is not providing a good fit to the data. Another definition of R2 requires a consideration of what the model is attempting to explain. What the model is trying to do in effect is to explain variability of y about its mean value, This quantity, which is more specifically known as the unconditional mean of y, acts like a benchmark since, if the researcher had no model for y, he could do no worse than to regress y on a constant only. In fact, the coefficient estimate for this regression would be the mean of y. So, from the regression (4.36) the coefficient estimate will be the mean of y, i.e., The total variation across all observations of the dependent variable about its mean value is known as the total sum of squares, TSS, which is given by: (4.37) The TSS can be split into two parts: the part that has been explained by the model (known as the explained sum of squares, ESS) and the part that the model was not able to explain (the RSS). That is (4.38) (4.39) Recall also that the residual sum of squares can also be expressed as

since a residual for observation t is defined as the difference between the actual and fitted values for that observation. The goodness of fit statistic is given by the ratio of the explained sum of squares to the total sum of squares: (4.40) but since TSS = ESS + RSS, it is also possible to write 226

(4.41)

R2 must always lie between zero and one (provided that there is a constant term in the regression). This is intuitive from the correlation interpretation of R2 given above, but for another explanation, consider two extreme cases

In the first case, the model has not succeeded in explaining any of the variability of y about its mean value, and hence the residual and total sums of squares are equal. This would happen only where the estimated values of all of the coefficients were exactly zero. In the second case, the model has explained all of the variability of y about its mean value, which implies that the residual sum of squares will be zero. This would happen only in the case where all of the observation points lie exactly on the fitted line. Neither of these two extremes is likely in practice, of course, but they do show that R2 is bounded to lie between zero and one, with a higher R2 implying, everything else being equal, that the model fits the data better. To sum up, a simple way (but crude, as explained next) to tell whether the regression line fits the data well is to look at the value of R2. A value of R2 close to 1 indicates that the model explains nearly all of the variability of the dependent variable about its mean value, while a value close to zero indicates that the model fits the data poorly. The two extreme cases, where R2 = 0 and R2 = 1, are indicated in Figures 4.1 and 4.2 in the context of a simple bivariate regression.

227

Figure 4.1 R2 = 0 demonstrated by a flat estimated line, i.e., a zero slope

coefficient

Figure 4.2 R2 = 1 when all data points lie exactly on the estimated line

4.7.2 Problems with R2 as a Goodness of Fit Measure R2 is simple to calculate, intuitive to understand, and provides a broad indication of the fit of the model to the data. However, there are a number of problems with R2 as a goodness of fit measure: (1) R2 is defined in terms of variation about the mean of y so that if a model is reparameterised (rearranged) and the dependent variable changes, R2 will change, even if the second model was a simple rearrangement of the first, with identical RSS. Thus it is not sensible 228

to compare the value of R2 across models with different dependent variables. (2) R2 never falls if more regressors are added to the regression. For example, consider the following two models: (4.42) (4.43) R2 will always be at least as high for regression 2 relative to regression 1. The R2 from regression 2 would be exactly the same as that for regression 1 only if the estimated value of the coefficient on the new variable were exactly zero, i.e., In practice, will always be non-zero, even if not significantly so, and thus in practice R2 always rises as more variables are added to a model. This feature of R2 essentially makes it impossible to use as a determinant of whether a given variable should be present in the model or not. (3) R2 can take values of 0.9 or higher for time-series regressions, and hence it is not good at discriminating between models, since a wide array of models will frequently have broadly similar (and high) values of R2.

4.7.3 Adjusted R2 In order to get around the second of these three problems, a modification to R2 is often made which takes into account the loss of degrees of freedom associated with adding extra variables. This is known as or 2 adjusted R , which is defined as (4.44) So if an extra regressor (variable) is added to the model, k increases and unless R2 increases by a more than off-setting amount, will actually fall. Hence can be used as a decision-making tool for determining whether a given variable should be included in a regression model or not, with the rule being: include the variable if rises and do not include it if falls. However, there are still problems with the maximisation of as criterion for model selection, and principal among these is that it is a ‘soft’ 229

rule, implying that by following it, the researcher will typically end up with a large model, containing a lot of marginally significant or insignificant variables. Also, while R2 must be at least zero if an intercept is included in the regression, its adjusted counterpart may take negative values, even with an intercept in the regression, if the model fits the data very poorly. To provide a couple of illustrations, if we consider a hedging example regression of spot returns on futures returns giving an R2 value of 0.99, this would indicate that almost all of the variation in spot returns is explained by the futures returns. However, regressions in the context of the CAPM time-series regression of excess stock returns on excess market returns often fit less well – suppose, for example, the value is R2 = 0.35 (i.e., about 35%), the conclusion would be that for that stock and sample period, around a third of the monthly movement in the excess returns can be attributed to movements in the market as a whole, as measured by the S&P500. There now follows another case study of the application of the OLS method of regression estimation, including interpretation of t-ratios and R2.

4.8 Hedonic Pricing Models An application of econometric techniques where the coefficients have a particularly intuitively appealing interpretation is in the area of hedonic pricing models. Hedonic models are used to value real assets, especially housing, and view the asset as representing a bundle of characteristics, each of which gives either utility or disutility to its consumer. Hedonic models are often used to produce appraisals or valuations of properties, given their characteristics (e.g., size of dwelling, number of bedrooms, location, number of bathrooms, etc.). In these models, the coefficient estimates represent ‘prices of the characteristics’. One such application of a hedonic pricing model is given by Des Des Rosiers and Thériault (1996), who consider the effect of various amenities on rental values for buildings and apartments in five sub-markets in the Quebec area of Canada. After accounting for the effect of ‘contractspecific’ features which will affect rental values (such as whether furnishings, lighting, or hot water are included in the rental price), they arrive at a model where the rental value in Canadian dollars per month (the dependent variable) is a function of nine to fourteen variables (depending 230

on the area under consideration). The paper employs 1990 data for the Quebec City region, and there are 13,378 observations. The twelve explanatory variables are: LnAGE

log of the apparent age of the property

NBROOMS

number of bedrooms

AREABYRM

area per room (in square metres)

ELEVATOR

a dummy variable = 1 if the building has an elevator; 0 otherwise

BASEMENT

a dummy variable = 1 if the unit is located in a basement; 0 otherwise

OUTPARK

number of outdoor parking spaces

INDPARK

number of indoor parking spaces

NOLEASE

a dummy variable = 1 if the unit has no lease attached to it; 0 otherwise

LnDISTCBD

log of the distance in kilometres to the central business district (CBD)

SINGLPAR

percentage of single-parent families in the area where the building stands

DSHOPCNTR

distance in kilometres to the nearest shopping centre

VACDIFF1

vacancy difference between the building and the census figure

This list includes several dummy variables, including ELEVATOR, BASEMENT, OUTPARK, INDPARK, NOLEASE. The interpretation of the coefficients on these dummies will be discussed when considering the output below. Des Rosiers and Thériault (1996) report several 231

specifications for five different regions, and they present results for the model with variables as discussed here in their Exhibit 4, which is adapted and reported here as Table 4.1. Table 4.1 Hedonic model of rental values in Quebec City, 1990. Dependent variable: Canadian dollars per month

Notes: Adjusted R2 = 0.651; regression F-statistic = 2082.27. Source: Des Rosiers and Thériault (1996). Reprinted with the permission of the American Real Estate Society.

The adjusted R2 value indicates that 65% of the total variability of rental prices about their mean value is explained by the model. For a crosssectional regression, this is quite high. Also, all variables are significant at the 0.01% level or lower and consequently, the regression F-statistic rejects very strongly the null hypothesis that all coefficient values on explanatory variables are zero. Note that there is a relationship between the regression F-statistic and R2, as shown in Box 4.1. BOX 4.1 The relationship between the regression F-statistic and R2 There is a particular relationship between a regression’s R2 value and 232

the regression F-statistic. Recall that the regression F-statistic tests the null hypothesis that all of the regression slope parameters are simultaneously zero. Let us call the residual sum of squares for the unrestricted regression including all of the explanatory variables RSS, while the restricted regression will simply be one of yt on a constant (4.45) Since there are no slope parameters in this model, none of the variability of yt about its mean value would have been explained. Thus the residual sum of squares for equation (4.45) will actually be the total sum of squares of yt, TSS. We could write the usual F-statistic formula for testing this null that all of the slope parameters are jointly zero as (4.46) In this case, the number of restrictions (‘m’) is equal to the number of slope parameters, k – 1. Recall that TSS – RSS = ESS and dividing the numerator and denominator of equation (4.46) by TSS, we obtain (4.47) Now the numerator of the left-hand part of equation (4.47) is R2, while the denominator is 1 – R2, so that the F-statistic can be written (4.48) This relationship between the F-statistic and R2 holds only for a test of this null hypothesis and not for any others. As stated above, one way to evaluate an econometric model is to determine whether it is consistent with theory. In this instance, no real theory is available, but instead there is a notion that each variable will affect rental values in a given direction. The actual signs of the coefficients can be compared with their expected values, given in the last column of Table 4.1 (as determined by this author). It can be seen that all coefficients 233

except two (the log of the distance to the CBD and the vacancy differential) have their predicted signs. It is argued by Des Rosiers and Thériault that the ‘distance to the CBD’ coefficient may be expected to have a positive sign since, while it is usually viewed as desirable to live close to a town centre, everything else being equal, in this instance most of the least desirable neighbourhoods are located towards the centre. The coefficient estimates themselves show the Canadian dollar rental price per month of each feature of the dwelling. To offer a few illustrations, the NBROOMS value of 48 (rounded) shows that, everything else being equal, one additional bedroom will lead to an average increase in the rental price of the property by $48 per month at 1990 prices. A basement coefficient of –16 suggests that an apartment located in a basement commands a rental $16 less than an identical apartment above ground. Finally the coefficients for parking suggest that on average each outdoor parking space adds $7 to the rent while each indoor parking space adds $74, and so on. The intercept shows, in theory, the rental that would be required of a property that had zero values on all the attributes. This case demonstrates, as stated previously, that the coefficient on the constant term often has little useful interpretation, as it would refer to a dwelling that has just been built, has no bedrooms each of zero size, no parking spaces, no lease, right in the CBD and shopping centre, etc. One limitation of such studies that is worth mentioning at this stage is their assumption that the implicit price of each characteristic is identical across types of property, and that these characteristics do not become saturated. In other words, it is implicitly assumed that if more and more bedrooms or allocated parking spaces are added to a dwelling indefinitely, the monthly rental price will rise each time by $48 and $7, respectively. This assumption is very unlikely to be upheld in practice, and will result in the estimated model being appropriate for only an ‘average’ dwelling. For example, an additional indoor parking space is likely to add far more value to a luxury apartment than a basic one. Similarly, the marginal value of an additional bedroom is likely to be bigger if the dwelling currently has one bedroom than if it already has ten. One potential remedy for this would be to use dummy variables with fixed effects in the regressions; see, for example, Chapter 11 for an explanation of these.

4.9 Tests of Non-Nested Hypotheses All of the hypothesis tests conducted thus far in this book have been in the context of ‘nested’ models. This means that, in each case, the test involved 234

imposing restrictions on the original model to arrive at a restricted formulation that would be a sub-set of, or nested within, the original specification. However, it is sometimes of interest to compare between non-nested models. For example, suppose that there are two researchers working independently, each with a separate financial theory for explaining the variation in some variable, yt. The models selected by the researchers respectively could be (4.49) (4.50) where ut and vt are iid error terms. Model (4.49) includes variable x2 but not x3, while model (4.50) includes x3 but not x2. In this case, neither model can be viewed as a restriction of the other, so how then can the two models be compared as to which better represents the data, yt? Given the discussion in Section 4.7, an obvious answer would be to compare the values of R2 or adjusted R2 between the models. Either would be equally applicable in this case since the two specifications have the same number of RHS variables. Adjusted R2 could be used even in cases where the number of variables was different across the two models, since it employs a penalty term that makes an allowance for the number of explanatory variables. However, adjusted R2 is based upon a particular penalty function (that is, T – k appears in a specific way in the formula). This form of penalty term may not necessarily be optimal. Also, given the statement above that adjusted R2 is a soft rule, it is likely on balance that use of it to choose between models will imply that models with more explanatory variables are favoured. Several other similar rules are available, each having more or less strict penalty terms; these are collectively known as ‘information criteria’. These are explained in some detail in Chapter 6, but suffice to say for now that a different strictness of the penalty term will in many cases lead to a different preferred model. An alternative approach to comparing between non-nested models would be to estimate an encompassing or hybrid model. In the case of equations (4.49) and (4.50), the relevant encompassing model would be (4.51)

235

where wt is an error term. Formulation (4.51) contains both equations (4.49) and (4.50) as special cases when γ3 and γ2 are zero, respectively. Therefore, a test for the best model would be conducted via an examination of the significances of γ2 and γ3 in model (4.51). There will be four possible outcomes (Box 4.2). BOX 4.2 Selecting between models (1) γ2 is statistically significant but γ3 is not. In this case, model (4.51) collapses to model (4.49), and the latter is the preferred model. (2) γ3 is statistically significant but γ2 is not. In this case, model (4.51) collapses to model (4.50), and the latter is the preferred model. (3) γ2 and γ3 are both statistically significant. This would imply that both x2 and x3 have incremental explanatory power for y, in which case both variables should be retained. Models (4.49) and (4.50) are both ditched and model (4.51) is the preferred model. (4) Neither γ2 nor γ3 are statistically significant. In this case, none of the models can be dropped, and some other method for choosing between them must be employed. However, there are several limitations to the use of encompassing regressions to select between non-nested models. Most importantly, even if models (4.49) and (4.50) have a strong theoretical basis for including the RHS variables that they do, the hybrid model may be meaningless. For example, it could be the case that financial theory suggests that y could either follow model (4.49) or model (4.50), but model (4.51) is implausible. Also, if the competing explanatory variables x2 and x3 are highly related (i.e., they are near collinear), it could be the case that if they are both included, neither γ2 nor γ3 is statistically significant, while each is significant in their separate regressions (4.49) and (4.50); see the section on multicollinearity in Chapter 5. An alternative approach is via the J-encompassing test due to Davidson and MacKinnon (1981). Interested readers are referred to their work or to Gujarati (2003, pp. 533–6) for further details. 236

4.10 Quantile Regression 4.10.1 Background and Motivation Standard regression approaches effectively model the (conditional) mean of the dependent variable – that is, they capture the average value of y given the average values of all of the explanatory variables. We could of course calculate from the fitted regression line the value that y would take for any values of the explanatory variables, but this would essentially be an extrapolation of the behaviour of the relationship between y and x at the mean to the remainder of the data. As a motivational example of why this approach will often be suboptimal, suppose that it is of interest to capture the cross-sectional relationship across countries between the degree of regulation of banks and gross domestic product (GDP). Starting from a very low level of regulation (or no regulation), an increase in regulation is likely to encourage a rise in economic activity as the banking system functions better as a result of more trust and stability in the financial environment. However, there is likely to come a point where further increasing the amount of regulation may impede economic growth by stifling innovation and the responsiveness of the banking sector to the needs of the industries it serves. Thus we may think of there being a non-linear (∩-shaped) relationship between regulation and GDP growth, and estimating a standard linear regression model may lead to seriously misleading estimates of this relationship as it will ‘average’ the positive and negative effects from very low and very high regulation. Of course, in this situation it would be possible to include non-linear (i.e., polynomial) terms in the regression model (for example, squared, cubic, … terms of regulation in the equation). But quantile regressions, developed by Koenker and Bassett (1978), represent a more natural and flexible way to capture the complexities inherent in the relationship by estimating models for the conditional quantile functions. Quantile regressions can be conducted in both time-series and cross-sectional contexts, although the latter are more common. It is usually assumed that the dependent variable, often called the response variable in the literature on quantile regressions, is independently distributed and homoscedastic; these assumptions can of course be relaxed but at the cost of additional complexity. Quantile regressions represent a comprehensive way to analyse the relationships between a set of variables, and are far more robust to outliers and non-normality than OLS regressions, in the same 237

fashion that the median is often a better measure of average or ‘typical’ behaviour than the mean when the distribution is considerably skewed by a few large outliers. Quantile regression is a non-parametric technique since no distributional assumptions are required to optimally estimate the parameters. The notation and approaches commonly used in quantile regression modelling are different to those that we are familiar with in financial econometrics, and this probably limited the early take up of the technique, which was historically more widely used in other disciplines. Numerous applications in labour economics were developed, for example. However, the more recent availability of the techniques in econometric software packages and increased interest in modelling the ‘tail behaviour’ of series have spurred applications of quantile regression in finance. The most common use of the technique here is to value at risk modelling. This seems natural given that the models are based on estimating the quantile of a distribution of possible losses – see, for example, the study by Chernozhukov and Umantsev (2001) and the development of the CaViaR model by Engle and Manganelli (2004).1 Quantiles, denoted τ, refer to the position where an observation falls within an ordered series for y – for example, the median is the observation in the very middle; the (lower) tenth percentile is the value that places 10% of observations below it (and therefore 90% of observations above), and so on. More precisely, we can define the τ-th quantile, Q(τ), of a random variable y having cumulative distribution F(y) as (4.52) where inf refers to the infimum, or the ‘greatest lower bound’ which is the smallest value of y satisfying the inequality. By definition, quantiles must lie between zero and one. Quantile regressions take the concept of quantiles a stage further and effectively model the entire conditional distribution of y given the explanatory variables (rather than only the mean as is the case for OLS) – thus they examine their impact on not only the location and scale of the distribution of y, but also on the shape of the distribution as well. So we can determine how the explanatory variables affect the fifth or ninetieth percentiles of the distribution of y or its median and so on.

4.10.2 Estimation of Quantile Functions 238

In the same fashion as the OLS estimator finds the mean value that minimises the sum of the squared residuals, minimising the sum of the absolute values of the residuals will yield the median value. By definition, the absolute value function is symmetrical so that the median always has the same number of data points above it as below it. But if instead the absolute residuals are weighted differently depending on whether they are positive or negative, we can calculate the quantiles of the distribution. To estimate the τ-th quantile, we would set the weight on positive observations to τ, which is the quantile of interest, and that on negative observations to 1 – τ. We can select the quantiles of interest (or the software might do this for us), but common choices would be 0.05, 0.1, 0.25, 0.5, 0.75, 0.9, 0.95. The fit is not always good for values of τ too close to its limits of 0 and 1, so it is advisable to avoid such values. We could write the minimisation problem for a set of quantile regression parameters each element of which is a k × 1 vector, as (4.53) This equation makes it clear where the weighting enters into the optimisation. As above, for the median, τ = 0.5 and the weights are symmetric, but for all other quantiles they will be asymmetric. This optimisation problem can be solved using a linear programming representation via the simplex algorithm or it can be cast within the generalised method of moments (GMM) framework. As an alternative to quantile regression, it would be tempting to think of partitioning the data and running separate regressions on each of them – for example, dropping the top 90% of the observations on y and the corresponding data points for the xs, and running a regression on the remainder. However, this process, tantamount to truncating the dependent variable, would be wholly inappropriate and could lead to potentially severe sample selection biases of the sort discussed in Chapter 12 and highlighted by Heckman (1976). In fact, quantile regression does not partition the data – all observations are used in the estimation of the parameters for every quantile. It is quite useful to plot each of the estimated parameters, against the quantile, τ (from 0 to 1) so that we can see whether the estimates vary across the quantiles or are roughly constant. Sometimes ±2 standard error bars are also included on the plot, and these 239

tend to widen as the limits of τ are approached. Producing these standard errors for the quantile regression parameters is unfortunately more complex conceptually than estimating the parameters themselves and thus a discussion of these is beyond the scope of this book. Under some assumptions, Koenker (2005) demonstrates that the quantile regression parameters are asymptotically normally distributed. A number of approaches have been proposed for estimating the variance-covariance matrix of the parameters, including one based on a bootstrap – see Chapter 13 for a discussion of this.

4.10.3 An Application of Quantile Regression: Evaluating Fund Performance A study by Bassett and Chen (2001) performs a style attribution analysis for a mutual fund and, for comparison, the S&P500 index. In order to examine how a portfolio’s exposure to various styles varies with performance, they use a quantile regression approach. Effectively evaluating the performance of mutual fund managers is made difficult by the observation that certain investment styles – notably, value and small cap – yield higher returns on average than the equity market as a whole. In response to this, factor models such as those of Fama and French (1993) have been employed to remove the impact of these characteristics – see Chapter 14 for a detailed presentation of these models. The use of such models also ensures that fund manager skill in picking highly performing stocks is not confused with randomly investing within value and small cap styles that will outperform the market in the long run. For example, if a manager invests a relatively high proportion of his portfolio in small firms, we would expect to observe higher returns than average from this manager because of the firm size effect alone. Bassett and Chen (2001) conduct a style analysis in this spirit by regressing the returns of a fund on the returns of a large growth portfolio, the returns of a large value portfolio, the returns of a small growth portfolio, and the returns of a small value portfolio. These style portfolio returns are based on the Russell style indices. In this way, the parameter estimates on each of these style-mimicking portfolio returns will measure the extent to which the fund is exposed to that style. Thus we can determine the actual investment style of a fund without knowing anything about its holdings purely based on an analysis of its returns ex post and their relationships with the returns of style indices. Table 4.2 presents the results from a standard OLS regression and quintile regressions for τ = 0.1, 240

0.3, 0.5 (i.e., the median), 0.7 and 0.9. The data are observed over the five years to December 1997 and the standard errors are based on a bootstrapping procedure. Table 4.2 OLS and quantile regression results for the Magellan fund

Large growth Large value Small growth Small value Constant

OLS

Q(0.1)

Q(0.3)

Q(0.5)

Q(0.7)

Q(0.9)

0.14

0.35

0.19

0.01

0.12

0.01

(0.15)

(0.31)

(0.22)

(0.16)

(0.20)

(0.22)

0.69

0.31

0.75

0.83

0.85

0.82

(0.20)

(0.38)

(0.30)

(0.25)

(0.30)

(0.36)

0.21

–0.01

0.10

0.14

0.27

0.53

(0.11)

(0.15)

(0.16)

(0.17)

(0.17)

(0.15)

–0.03

0.31

0.08

0.07

–0.31

–0.51

(0.20)

(0.31)

(0.27)

(0.29)

(0.32)

(0.35)

–0.05

–1.90

–1.11

–0.30

0.89

2.31

(0.25)

(0.39)

(0.27)

(0.38)

(0.40)

(0.57)

Notes: Standard errors in parentheses. Source: Bassett and Chen (2001). Reprinted with the permission of Springer-Verlag.

Notice that the sum of the style parameters for a given regression is always one (except for rounding errors). To conserve space, I only present the results for the Magellan active fund and not those for the S&P – the latter exhibit very little variation in the estimates across the quantiles. The OLS results (column 2) show that the mean return has by far its biggest exposure to large value stocks (and this parameter estimate is also statistically significant), but it also exposed to small growth and, to a lesser extent, large growth stocks. It is of interest to compare the mean (OLS) results with those for the median, Q(0.5). The latter show much higher exposure to large value, less to small growth and none at all to large growth. 241

It is also of interest to examine the factor tilts as we move through the quantiles from left (Q(0.1)) to right (Q(0.9)). We can see that the loading on large growth monotonically falls from 0.31 at Q(0.1) to 0.01 at Q(0.9) while the loadings on large value and small growth substantially increase. The loading on small value falls from 0.31 at Q(0.1) to -0.51 at Q(0.9). A way to interpret (those of the current authors rather than those of Bassett and Chen) these results is to say that when the fund has historically performed poorly, this has resulted in equal amounts from its overweight exposure to large value and growth, and small growth. On the other hand, when it has historically performed well, this is a result of its exposure to large value and small growth but it was underweight small value stocks. Finally, it is obvious that the intercept (coefficient on the constant) estimates should be monotonically increasing from left to right since the quantile regression effectively sorts on average performance and the intercept can be interpreted as the performance expected if the fund had zero exposure to all of the styles. KEY CONCEPTS The key terms to be able to define and explain from this chapter are multiple regression model restricted regression residual sum of squares multiple hypothesis test R2 hedonic model data mining dummy variables variance-covariance matrix F-distribution total sum of squares non-nested hypotheses encompassing regression quantile regression qualitative data

Appendix 4.1 Mathematical Derivations of CLRM 242

Results Derivation of the OLS Coefficient Estimator in the Multiple Regression Context In the multiple regression context, in order to obtain the parameter estimates for β1, β2, …, βk, the RSS would be minimised with respect to all the elements of β. Now the residuals are expressed in a vector

(4A.1)

The RSS is still the relevant loss function, and would be given in a matrix notation by expression (4A.2)

(4A.2)

Denoting the vector of estimated parameters as write

it is also possible to

(4A.3) It turns out that is (1 × k) × (k × T) × (T × 1) = 1 × 1 and also that is (1 × T) × (T × k) × (k × 1) = 1 × 1, so in fact Thus equation (4A.3) can be written (4A.4) Differentiating this expression with respect to and setting it to zero in order to find the parameter values that minimise the residual sum of squares would yield (4A.5) This expression arises since the derivative of y′y is zero with respect to 243

and acts like a square of Rearranging expression (4A.5)

which is differentiated to

(4A.6) (4A.7) Pre-multiplying both sides of (4A.7) by the inverse of X′X (4A.8) Thus, the vector of OLS coefficient estimates for a set of k parameters is given by

(4A.9)

Derivation of the OLS Standard Error Estimator in the Multiple Regression Context The variance of a vector of random variables is given by the formula Since y = Xβ + u, it can also be stated, given (4A.9), that (4A.10) Expanding the parentheses (4A.11) (4A.12) Thus, it is possible to express the variance of as (4A.13) Cancelling the β terms in each set of parentheses (4A.14) 244

Expanding the parentheses on the RHS of (4A.14) gives (4A.15) (4A.16) Now E[uu′] is estimated by s2I, so that (4A.17) where I is a k × k identity matrix. Rearranging further (4A.18) The X′X and the last (X′X)–1 term cancel out to leave (4A.19) as the expression for the parameter variance–covariance matrix. This quantity, s2(X′X)–1, is known as the estimated variance–covariance matrix of the coefficients. The leading diagonal terms give the estimated coefficient variances while the off-diagonal terms give the estimated covariances between the parameter estimates. The variance of is the first diagonal element, the variance of is the second element on the leading diagonal, …, and the variance of is the kth diagonal element, etc. as discussed in the body of the chapter.

Appendix 4.2 A Brief Introduction to Factor Models and Principal Components Analysis Factor models are employed primarily as dimensionality reduction techniques in situations where we have a large number of closely related variables and where we wish to allow for the most important influences from all of these variables at the same time. Factor models decompose the structure of a set of series into factors that are common to all series and a proportion that is specific to each series (idiosyncratic variation). There are broadly two types of such models, which can be loosely characterised as either macroeconomic or mathematical factor models. The key distinction between the two is that the factors are observable for the former but are latent (unobservable) for the latter. Observable factor models include the 245

APT model of Ross (1976). The most common mathematical factor model is principal components analysis (PCA). PCA is a technique that may be useful where explanatory variables are closely related – for example, in the context of near multicollinearity. Specifically, if there are k explanatory variables in the regression model, PCA will transform them into k uncorrelated new variables. To elucidate, suppose that the original explanatory variables are denoted x1, x2, …, xk, and denote the principal components by p1, p2, …, pk. These principal components are independent linear combinations of the original data

(4A.20)

where αij are coefficients to be calculated, representing the coefficient on the jth explanatory variable in the ith principal component. These coefficients are also known as factor loadings. Note that there will be T observations on each principal component if there were T observations on each explanatory variable. It is also required that the sum of the squares of the coefficients for each component is one, i.e.

(4A.21)

This requirement could also be expressed using sigma notation (4A.22) Constructing the components is a purely mathematical exercise in constrained optimisation, and thus no assumption is made concerning the structure, distribution, or other properties of the variables. The principal components are derived in such a way that they are in descending order of importance. Although there are k principal components, the same as the number of explanatory variables, if there is 246

some collinearity between these original explanatory variables, it is likely that some of the (last few) principal components will account for so little of the variation that they can be discarded. However, if all of the original explanatory variables were already essentially uncorrelated, all of the components would be required, although in such a case there would have been little motivation for using PCA in the first place. The principal components can also be understood as the eigenvalues of (X′X), where X is the matrix of observations on the original variables. Thus thenumber of eigenvalues will be equal to the number of variables, k. If the ordered eigenvalues are denoted λi (i = 1, …, k), the ratio

gives the proportion of the total variation in the original data explained by the principal component i. Suppose that only the first r (0 < r < k) principal components are deemed sufficiently useful in explaining the variation of (X′X), and that they are to be retained, with the remaining k – r components being discarded. The regression finally estimated, after the principal components have been formed, would be one of y on the r principal components (4A.23) In this way, the principal components are argued to keep most of the important information contained in the original explanatory variables, but are orthogonal. This may be particularly useful for independent variables that are very closely related. The principal component estimates will be biased estimates, although they will be more efficient than the OLS estimators since redundant information has been removed. In fact, if the OLS estimator for the original regression of y on x is denoted it can be shown that (4A.24) where are the coefficient estimates for the principal components, and Pr is a matrix of the first r principal components. The principal component coefficient estimates are thus simply linear combinations of the original OLS estimates. 247

An Application of Principal Components to Interest Rates Many economic and financial models make use of interest rates in some form or another as independent variables. Researchers may wish to include interest rates on a large number of different assets in order to reflect the variety of investment opportunities open to investors. However, market interest rates could be argued to be not sufficiently independent of one another to make the inclusion of several interest rate series in an econometric model statistically sensible. One approach to examining this issue would be to use PCA on several related interest rate series to determine whether they did move independently of one another over some historical time period or not. Fase (1973) conducted such a study in the context of monthly Dutch market interest rates from January 1962 until December 1970 (108 months). Fase examined both ‘money market’ and ‘capital market’ rates, although only the money market results will be discussed here in the interests of brevity. The money market instruments investigated were Call money Three-month Treasury paper One-year Treasury paper Two-year Treasury paper Three-year Treasury paper Five-year Treasury paper Loans to local authorities: three-month Loans to local authorities: one-year Eurodollar deposits Netherlands Bank official discount rate. Prior to analysis, each series was standardised to have zero mean and unit variance by subtracting the mean and dividing by the standard deviation in each case. The three largest of the ten eigenvalues are given in Table 4A.1. Table 4A.1 Principal component ordered eigenvalues for Dutch interest rates, 1962–70

248

Source: Fase (1973). Reprinted with the permission of Elsevier.

The results in Table 4A.1 are presented for the whole period using the monthly data, for two monthly sub-samples, and for the whole period using data sampled quarterly instead of monthly. The results show clearly that the first principal component is sufficient to describe the common variation in these Dutch interest rate series. The first component is able to explain over 90% of the variation in all four cases, as given in the last row of Table 4A.1. Clearly, the estimated eigenvalues are fairly stable across the sample periods and are relatively invariant to the frequency of sampling of the data. The factor loadings (coefficient estimates) for the first two ordered components are given in Table 4A.2. Table 4A.2 Factor loadings of the first and second principal components for Dutch interest rates, 1962–70

Source: Fase (1973). Reprinted with the permission of Elsevier.

249

As Table 4A.2 shows, the loadings on each factor making up the first principal component are all positive. Since each series has been standardised to have zero mean and unit variance, the coefficients αj1 and αj2 can be interpreted as the correlations between the interest rate j and the first and second principal components, respectively. The factor loadings for each interest rate series on the first component are all very close to one. Fase (1973) therefore argues that the first component can be interpreted simply as an equally weighted combination of all of the market interest rates. The second component, which explains much less of the variability of the rates, shows a factor loading pattern of positive coefficients for the Treasury paper series and negative or almost zero values for the other series. Fase (1973) argues that this is owing to the characteristics of the Dutch Treasury instruments that they rarely change hands and have low transactions costs, and therefore have less sensitivity to general interest rate movements. Also, they are not subject to default risks in the same way as, for example Eurodollar deposits. Therefore, the second principal component is broadly interpreted as relating to default risk and transactions costs. Principal components can be useful in some circumstances, although the technique has limited applicability for the following reasons A change in the units of measurement of x will change the principal components. It is thus usual to transform all of the variables to have zero mean and unit variance prior to applying PCA. The principal components usually have no theoretical motivation or interpretation whatsoever. The r principal components retained from the original k are the ones that explain most of the variation in x, but these components might not be the most useful as explanations for y. SELF-STUDY QUESTIONS 1. By using examples from the relevant statistical tables, explain the relationship between the t- and the F-distributions. For questions 2–5, assume that the econometric model is of the form (4.54) 250

2. Which of the following hypotheses about the coefficients can be tested using a t-test? Which of them can be tested using an F-test? In each case, state the number of restrictions. (a) H0 : β3 = 2 (b) H0 : β3 + β4 = 1 (c) H0 : β3 + β4 = 1 and β5 = 1 (d) H0 : β2 = 0 and β3 = 0 and β4 = 0 and β5 = 0 (e) H0 : β2β3 = 1 3. Which of the above null hypotheses constitutes ‘THE’ regression F-statistic in the context of equation (4.54)? Why is this null hypothesis always of interest whatever the regression relationship under study? What exactly would constitute the alternative hypothesis in this case? 4. Which would you expect to be bigger – the unrestricted residual sum of squares or the restricted residual sum of squares, and why? 5. You decide to investigate the relationship given in the null hypothesis of Question 2, part (c). What would constitute the restricted regression? The regressions are carried out on a sample of 96 quarterly observations, and the residual sums of squares for the restricted and unrestricted regressions are 102.87 and 91.41, respectively. Perform the test. What is your conclusion? 6. You estimate a regression of the form given by equation (4.55) below in order to evaluate the effect of various firm-specific factors on the returns of a sample of firms. You run a crosssectional regression with 200 firms (4.55) where: ri is the percentage annual return for the stock Si is the size of firm i measured in terms of sales revenue MBi is the market to book ratio of the firm PEi is the price/earnings (P/E) ratio of the firm BETAi is the stock’s CAPM beta coefficient You obtain the following results (with standard errors in 251

parentheses) (4.56) Calculate the t-ratios. What do you conclude about the effect of each variable on the returns of the security? On the basis of your results, what variables would you consider deleting from the regression? If a stock’s beta increased from 1 to 1.2, what would be the expected effect on the stock’s return? Is the sign on beta as you would have expected? Explain your answers in each case. 7. A researcher estimates the following econometric models including a lagged dependent variable (4.57) (4.58) where ut and vt are iid disturbances. Will these models have the same value of (a) The residual sum of squares (RSS), (b) R2, (c) Adjusted R2? Explain your answers in each case. 8. A researcher estimates the following two econometric models (4.59) (4.60) where ut and vt are iid disturbances and x3t is an irrelevant variable which does not enter into the data generating process for yt. Will the value of (a) R2, (b) Adjusted R2, be higher for the second model than the first? Explain your answers. 9. What are the units of R2? 10. What are quantile regressions and why are they useful? 11. A researcher wishes to examine the link between the returns on two assets A and B in situations where the price of B is falling rapidly. To do this, he orders the data according to changes in the price of B and drops the top 80% of ordered observations. He then 252

runs a regression of the returns of A on the returns of B for the remaining lowest 20% of observations. Would this be a good way to proceed? Explain your answer.

1

For further reading on quantile regression, Koenker and Hallock (2001) represents a very accessible, albeit brief, introduction to quantile regressions and their applications. A more thorough treatment is given in the book by Koenker (2005).

253

5 Classical Linear Regression Model Assumptions and Diagnostic Tests

LEARNING OUTCOMES In this chapter, you will learn how to Describe the steps involved in testing regression residuals for heteroscedasticity and autocorrelation Explain the impact of heteroscedasticity or autocorrelation on the optimality of OLS parameter and standard error estimation Distinguish between the Durbin–Watson and Breusch–Godfrey tests for autocorrelation Highlight the advantages and disadvantages of dynamic models Test for whether the functional form of the model employed is appropriate Determine whether the residual distribution from a regression differs significantly from normality Investigate whether the model parameters are stable Appraise different philosophies of how to build an econometric model

5.1 Introduction Recall from Chapter 3 that five assumptions were made relating to the classical linear regression model (CLRM). These were required to show that the estimation technique, ordinary least squares (OLS), had a number of desirable properties, and also so that hypothesis tests regarding the 254

coefficient estimates could validly be conducted. Specifically, it was assumed that (1) E(ut) = 0 (2) var(ut) = σ2 < ∞ (3) cov(ui,uj) = 0 (4) cov(ut,xt) = 0 (5) ut ~ N(0, σ2) These assumptions will now be studied further, in particular looking at the following How can violations of the assumptions be detected? What are the most likely causes of the violations in practice? What are the consequences for the model if an assumption is violated but this fact is ignored and the researcher proceeds regardless? The answer to the last of these questions is that, in general, the model could encounter any combination of three problems The coefficient estimates are wrong The associated standard errors are wrong The distributions that were assumed for the test statistics are inappropriate. A pragmatic approach to ‘solving’ problems associated with the use of models where one or more of the assumptions is not supported by the data will then be adopted. Such solutions usually operate such that The assumptions are no longer violated, or The problems are side-stepped, so that alternative techniques are used which are still valid.

5.2 Statistical Distributions for Diagnostic Tests The text below discusses various regression diagnostic (misspecification) tests that are based on the calculation of a test statistic. These tests can be constructed in several ways, and the precise approach to constructing the test statistic will determine the distribution that the test statistic is assumed 255

to follow. Two particular approaches are in common usage and their results are given by the statistical packages: the Lagrange Multiplier (LM) test and the Wald test. Further details concerning these procedures are given in Chapter 9. For now, all that readers require to know is that LM test statistics in the context of the diagnostic tests presented here follow a χ2 distribution with degrees of freedom equal to the number of restrictions placed on the model, and denoted m. The Wald version of the test follows an F-distribution with (m, T − k) degrees of freedom. Asymptotically, these two tests are equivalent, although their results will differ somewhat in small samples. They are equivalent as the sample size increases towards infinity since there is a direct relationship between the χ2- and Fdistributions. Asymptotically, an F-variate will tend towards a χ2 variate divided by its degrees of freedom

Computer packages typically present results using both approaches, although only one of the two will be illustrated for each test below. They will usually give the same conclusion, although if they do not, the Fversion is usually considered preferable for finite samples, since it is sensitive to sample size (one of its degrees of freedom parameters depends on sample size) in a way that the χ2-version is not.

5.3 Assumption (1): E(ut) = 0 The first assumption required is that the average value of the errors is zero. In fact, if a constant term is included in the regression equation, this assumption will never be violated. But what if financial theory suggests that, for a particular application, there should be no intercept so that the regression line is forced through the origin? If the regression did not include an intercept, and the average value of the errors was non-zero, several undesirable consequences could arise. First, R2, defined as ESS/TSS can be negative, implying that the sample average, ‘explains’ more of the variation in y than the explanatory variables. Second, and more fundamentally, a regression with no intercept parameter could lead to potentially severe biases in the slope coefficient estimates. To see this, consider Figure 5.1.

256

Figure 5.1 Effect of no intercept on a regression line

The solid line shows the regression estimated including a constant term, while the dotted line shows the effect of suppressing (i.e., setting to zero) the constant term. The effect is that the estimated line in this case is forced through the origin, so that the estimate of the slope coefficient is biased. Additionally, R2 and are usually meaningless in such a context. This arises since the mean value of the dependent variable, will not be equal to the mean of the fitted values from the model, i.e., the mean of if there is no constant in the regression.

5.4 Assumption (2): var(ut) = σ2 < ∞ It has been assumed thus far that the variance of the errors is constant, σ2 – this is known as the assumption of homoscedasticity. If the errors do not have a constant variance, they are said to be heteroscedastic. To consider one illustration of heteroscedasticity, suppose that a regression had been estimated and the residuals, have been calculated and then plotted against one of the explanatory variables, x2t, as shown in Figure 5.2.

257

Figure 5.2 Graphical illustration of heteroscedasticity

It is clearly evident that the errors in figure 5.2 are heteroscedastic – that is, although their mean value is roughly constant, their variance is increasing systematically with x2t.

5.4.1 Detection of Heteroscedasticity How can one tell whether the errors are heteroscedastic or not? It is possible to use a graphical method as above, but unfortunately one rarely knows the cause or the form of the heteroscedasticity, so that a plot is likely to reveal nothing. For example, if the variance of the errors was an increasing function of x3t, and the researcher had plotted the residuals against x2t, he would be unlikely to see any pattern and would thus wrongly conclude that the errors had constant variance. It is also possible that the variance of the errors changes over time rather than systematically with one of the explanatory variables; this phenomenon is known as ‘ARCH’ and is described in Chapter 9. Fortunately, there are a number of formal statistical tests for heteroscedasticity, and one of the simplest such methods is the Goldfeld– Quandt (1965) test. Their approach is based on splitting the total sample of length T into two sub-samples of length T1 and T2. The regression model is estimated on each sub-sample and the two residual variances are calculated as and respectively. The null hypothesis is that the variances of the disturbances are equal, which can be written 258

against a two-sided alternative. The test statistic, denoted GQ, is simply the ratio of the two residual variances where the larger of the two variances must be placed in the numerator (i.e., is the higher sample variance for the sample with length T1, even if it comes from the second sub-sample) (5.1) The test statistic is distributed as an F(T1 − k, T2 − k) under the null hypothesis, and the null of a constant variance is rejected if the test statistic exceeds the critical value. The GQ test is simple to construct but its conclusions may be contingent upon a particular, and probably arbitrary, choice of where to split the sample. Clearly, the test is likely to be more powerful when this choice is made on theoretical grounds – for example, before and after a major structural event. Suppose that it is thought that the variance of the disturbances is related to some observable variable zt (which may or may not be one of the regressors). A better way to perform the test would be to order the sample according to values of zt (rather than through time) and then to split the reordered sample into T1 and T2. An alternative method that is sometimes used to sharpen the inferences from the test and to increase its power is to omit some of the observations from the centre of the sample so as to introduce a degree of separation between the two sub-samples. A further popular test is White’s (1980) general test for heteroscedasticity. The test is particularly useful because it makes few assumptions about the likely form of the heteroscedasticity. The test is carried out as in Box 5.1. BOX 5.1 Conducting White’s test (1) Assume that the regression model estimated is of the standard linear form, e.g. (5.2) To test var(ut) = σ2, estimate the model above, obtaining the residuals, 259

(2) Then run the auxiliary regression (5.3) where vt is a normally distributed disturbance term independent of ut. This regression is of the squared residuals on a constant, the original explanatory variables, the squares of the explanatory variables and their cross-products. To see why the squared residuals are the quantity of interest, recall that for a random variable ut, the variance can be written (5.4) Under the assumption that E(ut) = 0, the second part of the RHS of this expression disappears (5.5) Once again, it is not possible to know the squares of the population disturbances, so their sample counterparts, the squared residuals, are used instead. The reason that the auxiliary regression takes this form is that it is desirable to investigate whether the variance of the residuals (embodied in ) varies systematically with any known variables relevant to the model. Relevant variables will include the original explanatory variables, their squared values and their cross-products. Note also that this regression should include a constant term, even if the original regression did not. This is as a result of the fact that will always have a non-zero mean, even if has a zero mean. (3) Given the auxiliary regression, as stated above, the test can be conducted using two different approaches. First, it is possible to use the F-test framework described in Chapter 4. This would involve estimating model (5.3) as the unrestricted regression and then running a restricted regression of on a constant only. The RSS from each specification would then be used as inputs to the standard F-test formula. With many diagnostic tests, an alternative approach can be adopted that does not require the estimation of a second 260

(restricted) regression. This approach is known as a Lagrange Multiplier (LM) test, which centres around the value of R2 for the auxiliary regression. If one or more coefficients in model (5.3) is statistically significant, the value of R2 for that equation will be relatively high, while if none of the variables is significant, R2 will be relatively low. The LM test would thus operate by obtaining R2 from the auxiliary regression and multiplying it by the number of observations, T. It can be shown that

where m is the number of regressors in the auxiliary regression (excluding the constant term), equivalent to the number of restrictions that would have to be placed under the F-test approach. (4) The test is one of the joint null hypothesis that α2 = 0, and α3 = 0, and α4 = 0, and α5 = 0, and α6 = 0. For the LM test, if the χ2test statistic from step (3) is greater than the corresponding value from the statistical table then reject the null hypothesis that the errors are homoscedastic.

5.4.2 Consequences of Using OLS in the Presence of Heteroscedasticity What happens if the errors are heteroscedastic, but this fact is ignored and the researcher proceeds with estimation and inference? In this case, OLS estimators will still give unbiased (and also consistent) coefficient estimates, but they are no longer best linear unbiased estimators (BLUE) – that is, they no longer have the minimum variance among the class of unbiased estimators. The reason is that the error variance, σ2, plays no part in the proof that the OLS estimator is consistent and unbiased, but σ2 does appear in the formulae for the coefficient variances. If the errors are heteroscedastic, the formulae presented for the coefficient standard errors no longer hold. For a very accessible algebraic treatment of the consequences of heteroscedasticity, see Hill, Griffiths and Judge (1997, pp. 217–18). EXAMPLE 5.1 261

Suppose that the model (5.2) above has been estimated using 120 observations, and the R2 from the auxiliary regression (5.3) is 0.234. The test statistic will be given by TR2 = 120 × 0.234 = 28.8, which will follow a χ2(5) under the null hypothesis. The 5% critical value from the χ2 table is 11.07. The test statistic is therefore more than the critical value and hence the null hypothesis is rejected. It would be concluded that there is significant evidence of heteroscedasticity, so that it would not be plausible to assume that the variance of the errors is constant in this case. So, the upshot is that if OLS is still used in the presence of heteroscedasticity, the standard errors could be wrong and hence any inferences made could be misleading. In general, the OLS standard errors will be too large for the intercept when the errors are heteroscedastic. The effect of heteroscedasticity on the slope standard errors will depend on its form. For example, if the variance of the errors is positively related to the square of an explanatory variable (which is often the case in practice), the OLS standard error for the slope will be too low. On the other hand, the OLS slope standard errors will be too big when the variance of the errors is inversely related to an explanatory variable.

5.4.3 Dealing with Heteroscedasticity If the form (i.e., the cause) of the heteroscedasticity is known, then an alternative estimation method which takes this into account can be used. One possibility is called generalised least squares (GLS). For example, suppose that the error variance was related to zt by the expression (5.6) All that would be required to remove the heteroscedasticity would be to divide the regression equation through by zt (5.7) where

is an error term. for known z.

Now, if 262

Therefore, the disturbances from equation (5.7) will be homoscedastic. Note that this latter regression does not include a constant since β1 is multiplied by (1/zt). GLS can be viewed as OLS applied to transformed data that satisfy the OLS assumptions. GLS is also known as weighted least squares (WLS), since under GLS a weighted sum of the squared residuals is minimised, whereas under OLS it is an unweighted sum. However, researchers are typically unsure of the exact cause of the heteroscedasticity, and hence this technique is usually infeasible in practice. Two other possible ‘solutions’ for heteroscedasticity are shown in Box 5.2. BOX 5.2 ‘Solutions’ for Heteroscedasticity (1) Transforming the variables into logs or reducing by some other measure of ‘size’. It is sometimes said that a log transform is appropriate when its standard deviation is proportional to its mean. This has the effect of rescaling the data to ‘pull in’ extreme observations. The regression would then be conducted upon the natural logarithms or the transformed data. Taking logarithms also has the effect of making a multiplicative model, such as the exponential regression model discussed previously (with a multiplicative error term), into an additive one. However, logarithms of a variable cannot be taken in situations where the variable can take on zero or negative values, for the log will not be defined in such cases. (2) Using heteroscedasticity-consistent standard error estimates. Most standard econometrics software packages have an option (usually called something like ‘robust’) that allows the user to employ standard error estimates that have been modified to account for the heteroscedasticity following White (1980). The effect of using the correction is that, if the variance of the errors is positively related to the square of an explanatory variable, the standard errors for the slope coefficients are increased relative to the usual OLS standard errors, which would make hypothesis testing more ‘conservative’, so that more evidence would be required against the null hypothesis before it would be rejected. Examples of tests for heteroscedasticity in the context of the single index market model are given in Fabozzi and Francis (1980). Their results 263

are strongly suggestive of the presence of heteroscedasticity, and they examine various factors that may constitute the form of the heteroscedasticity.

5.5 Assumption (3): cov(ui, uj) = 0 for i ≠ j Assumption 3 that is made of the CLRM’s disturbance terms is that the covariance between the error terms over time (or cross-sectionally, for that type of data) is zero. In other words, it is assumed that the errors are uncorrelated with one another. If the errors are not uncorrelated with one another, it would be stated that they are ‘autocorrelated’ or that they are ‘serially correlated’. A test of this assumption is therefore required. Again, the population disturbances cannot be observed, so tests for autocorrelation are conducted on the residuals, Before one can proceed to see how formal tests for autocorrelation are formulated, the concept of the lagged value of a variable needs to be defined.

5.5.1 The Concept of a Lagged Value The lagged value of a variable (which may be yt, xt, or ut) is simply the value that the variable took during a previous period. So for example, the value of yt lagged one period, written yt−1, can be constructed by shifting all of the observations forward one period in a spreadsheet, as illustrated in Table 5.1. Table 5.1 Constructing a series of lagged values and first differences

264

So, the value in the 2006M10 row and the yt−1 column shows the value that yt took in the previous period, 2006M09, which was 0.8. The last column in Table 5.1 shows another quantity relating to y, namely the ‘first difference’. The first difference of y, also known as the change in y, and denoted Δyt, is calculated as the difference between the values of y in this period and in the previous period. This is calculated as (5.8) Note that when one-period lags or first differences of a variable are constructed, the first observation is lost. Thus a regression of Δyt using the above data would begin with the October 2006 data point. It is also possible to produce two-period lags, three-period lags and so on. These would be accomplished in the obvious way.

5.5.2 Graphical Tests for Autocorrelation In order to test for autocorrelation, it is necessary to investigate whether any relationships exist between the current value of and any of its previous values, The first step is to consider possible relationships between the current residual and the immediately previous one, via a graphical exploration. Thus is plotted against and is plotted over time. Some stereotypical patterns that may be found in the residuals are discussed below. 265

Figures 5.3 and 5.4 show positive autocorrelation in the residuals, which is indicated by a cyclical residual plot over time. This case is known as positive autocorrelation since on average if the residual at time t − 1 is positive, the residual at time t is likely to be also positive; similarly, if the residual at t − 1 is negative, the residual at t is also likely to be negative. Figure 5.3 shows that most of the dots representing observations are in the first and third quadrants, while Figure 5.4 shows that a positively autocorrelated series of residuals will not cross the time-axis very frequently.

Figure 5.3 Plot of

against

showing positive autocorrelation

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Figure 5.4 Plot of

over time, showing positive autocorrelation

Figures 5.5 and 5.6 show negative autocorrelation, indicated by an alternating pattern in the residuals. This case is known as negative autocorrelation since on average if the residual at time t − 1 is positive, the residual at time t is likely to be negative; similarly, if the residual at t − 1 is negative, the residual at t is likely to be positive. Figure 5.5 shows that most of the dots are in the second and fourth quadrants, while Figure 5.6 shows that a negatively autocorrelated series of residuals will cross the time-axis more frequently than if they were distributed randomly.

267

Figure 5.5 Plot of

against

showing negative autocorrelation

Figure 5.6 Plot of

over time, showing negative autocorrelation

Finally, Figures 5.7 and 5.8 show no pattern in the residuals at all: this is what is desirable to see. In the plot of against Figure 5.7), the points are randomly spread across all four quadrants, and the time-series plot of the residuals (Figure 5.8) does not cross the x-axis either too frequently or too little.

268

Figure 5.7 Plot of

against

showing no autocorrelation

Figure 5.8 Plot of

over time, showing no autocorrelation

5.5.3 Detecting Autocorrelation: The Durbin–Watson Test Of course, a first step in testing whether the residual series from an estimated model are autocorrelated would be to plot the residuals as above, looking for any patterns. Graphical methods may be difficult to interpret in practice, however, and hence a formal statistical test should also be applied. The simplest test is due to Durbin and Watson (1951). 269

Durbin–Watson (DW) is a test for first order autocorrelation – i.e., it tests only for a relationship between an error and its immediately previous value. One way to motivate the test and to interpret the test statistic would be in the context of a regression of the time t error on its previous value (5.9) where hypotheses

The DW test statistic has as its null and alternative

Thus, under the null hypothesis, the errors at time t − 1 and t are independent of one another, and if this null were rejected, it would be concluded that there was evidence of a relationship between successive residuals. In fact, it is not necessary to run the regression given by equation (5.9) since the test statistic can be calculated using quantities that are already available after the first regression has been run

(5.10)

The denominator of the test statistic is simply (the number of observations −1) × the variance of the residuals. This arises since if the average of the residuals is zero

so that

The numerator ‘compares’ the values of the error at times t − 1 and t. If there is positive autocorrelation in the errors, this difference in the numerator will be relatively small, while if there is negative autocorrelation, with the sign of the error changing very frequently, the numerator will be relatively large. No autocorrelation would result in a value for the numerator between small and large. 270

It is also possible to express the DW statistic as an approximate function of the estimated value of ρ (5.11) where is the estimated correlation coefficient that would have been obtained from an estimation of equation (5.9). To see why this is the case, consider that the numerator of equation (5.10) can be written as the parts of a quadratic (5.12) Consider now the composition of the first two summations on the RHS of equation (5.12). The first of these is

while the second is

Thus, the only difference between them is that they differ in the first and last terms in the summation, so

contains

but not

while

contains but not As the sample size, T, increases towards infinity, the difference between these two will become negligible. Hence, the expression in equation (5.12), the numerator of equation (5.10), is approximately

271

Replacing the numerator of equation (5.10) with this expression leads to

(5.13)

The covariance between ut and ut−1 can be written as E[(ut − E(ut))(ut−1 − E(ut−1))]. Under the assumption that E(ut) = 0 (and therefore that E(ut−1) = 0), the covariance will be E[ut ut−1]. For the sample residuals, this covariance will be evaluated as

Thus, the sum in the numerator of the expression on the right of equation (5.13) can be seen as T − 1 times the covariance between and while the sum in the denominator of the expression on the right of equation (5.13) can be seen from the previous exposition as T − 1 times the variance of Thus, it is possible to write

(5.14)

so that the DW test statistic is approximately equal to Since is a correlation, it implies that That is, is bounded to lie between −1 and +1. Substituting in these limits for to calculate DW from equation (5.11) would give the corresponding limits for DW as 0 ≤ DW ≤ 4. Consider now the implication of DW taking one of three important values (0, 2, and 4): DW = 2 This is the case where there is no autocorrelation in the residuals. So roughly speaking, the null hypothesis would not be rejected if DW is near 2 → i.e., there is little evidence of autocorrelation. DW = 0 This corresponds to the case where there is perfect positive autocorrelation in the residuals. DW = 4 This corresponds to the case where there is perfect 272

negative autocorrelation in the residuals. The DW test does not follow a standard statistical distribution such as a t, F, or χ2. DW has 2 critical values: an upper critical value (dU) and a lower critical value (dL), and there is also an intermediate region where the null hypothesis of no autocorrelation can neither be rejected nor not rejected! The rejection, non-rejection and inconclusive regions are shown on the number line in Figure 5.9.

Figure 5.9 Rejection and non-rejection regions for DW test

So, to reiterate, the null hypothesis is rejected and the existence of positive autocorrelation presumed if DW is less than the lower critical value; the null hypothesis is rejected and the existence of negative autocorrelation presumed if DW is greater than 4 minus the lower critical value; the null hypothesis is not rejected and no significant residual autocorrelation is presumed if DW is between the upper and 4 minus the upper limits. EXAMPLE 5.2 A researcher wishes to test for first order serial correlation in the residuals from a linear regression. The DW test statistic value is 0.86. There are eighty quarterly observations in the regression, which is of the form (5.15) The relevant critical values for the test (see Table A2.6 in the appendix of statistical distributions at the end of this book), are dL = 1.42, dU = 1.57, so 4 − dU = 2.43 and 4 − dL = 2.58. The test statistic is clearly lower than the lower critical value and hence the null hypothesis of no autocorrelation is rejected and it would be concluded that the residuals from the model appear to be positively autocorrelated.

273

5.5.4 Conditions Which Must be Fulfilled for DW to be a Valid Test In order for the DW test to be valid for application, three conditions must be fulfilled (Box 5.3). BOX 5.3 Conditions for DW to be a valid test (1) There must be a constant term in the regression (2) The regressors must be non-stochastic – as assumption (4) of the CLRM (see Chapter 7) (3) There must be no lags of dependent variable (see Section 5.5.8) in the regression.

If the test were used in the presence of lags of the dependent variable or otherwise stochastic regressors, the test statistic would be biased towards 2, suggesting that in some instances the null hypothesis of no autocorrelation would not be rejected when it should be.

5.5.5 Another Test for Autocorrelation: The Breusch–Godfrey Test Recall that DW is a test only of whether consecutive errors are related to one another. So, not only can the DW test not be applied if a certain set of circumstances are not fulfilled, there will also be many forms of residual autocorrelation that DW cannot detect. For example, if but DW as defined above will not find any autocorrelation. One possible solution would be to replace in equation (5.10) with However, pairwise examinations of the correlations will be tedious in practice and is not coded in econometrics software packages, which have been programmed to construct DW using only a one-period lag. In addition, the approximation in equation (5.11) will deteriorate as the difference between the two time indices increases. Consequently, the critical values should also be modified somewhat in these cases. Therefore, it is desirable to examine a joint test for autocorrelation that will allow examination of the relationship between and several of its lagged values at the same time. The Breusch–Godfrey test is a more general test for autocorrelation up to the rth order. The model for the errors 274

under this test is (5.16) The null and alternative hypotheses are:

So, under the null hypothesis, the current error is not related to any of its r previous values. The test is carried out as in Box 5.4. BOX 5.4 Conducting a Breusch–Godfrey test (1) Estimate the linear regression using OLS and obtain the residuals, (2) Regress on all of the regressors from stage 1 (the xs) plus the regression will thus be (5.17) Obtain R2 from this auxiliary regression (3) Letting T denote the number of observations, the test statistic is given by

Note that (T − r) pre-multiplies R2 in the test for autocorrelation rather than T (as was the case for the heteroscedasticity test). This arises because the first r observations will effectively have been lost from the sample in order to obtain the r lags used in the test regression, leaving (T − r) observations from which to estimate the auxiliary regression. If the test statistic exceeds the critical value from the chi-squared statistical tables, reject the null hypothesis of no autocorrelation. As with any joint test, only one part of the null hypothesis has to be rejected to lead to rejection of the hypothesis as a whole. So the error at time t has to be significantly related only to one of its previous r values in the sample for the null of no autocorrelation to be rejected. The test is more general than the DW test, and can be applied in a wider variety of circumstances since it does not 275

impose the DW restrictions on the format of the first stage regression. One potential difficulty with Breusch–Godfrey, however, is in determining an appropriate value of r, the number of lags of the residuals, to use in computing the test. There is no obvious answer to this, so it is typical to experiment with a range of values, and also to use the frequency of the data to decide. So, for example, if the data are monthly or quarterly, set r equal to 12 or 4, respectively. The argument would then be that errors at any given time would be expected to be related only to those errors in the previous year. Obviously, if the model is statistically adequate, no evidence of autocorrelation should be found in the residuals whatever value of r is chosen.

5.5.6 Consequences of Ignoring Autocorrelation if it is Present In fact, the consequences of ignoring autocorrelation when it is present are similar to those of ignoring heteroscedasticity. The coefficient estimates derived using OLS are still unbiased, but they are inefficient, i.e., they are not BLUE, even at large sample sizes, so that the standard error estimates could be wrong. There thus exists the possibility that the wrong inferences could be made about whether a variable is or is not an important determinant of variations in y. In the case of positive serial correlation in the residuals, the OLS standard error estimates will be biased downwards relative to the true standard errors. That is, OLS will understate their true variability. This would lead to an increase in the probability of type I error – that is, a tendency to reject the null hypothesis sometimes when it is correct. Furthermore, R2 is likely to be inflated relative to its ‘correct’ value if autocorrelation is present but ignored, since residual autocorrelation will lead to an underestimate of the true error variance (for positive autocorrelation).

5.5.7 Dealing with Autocorrelation If the form of the autocorrelation is known, it would be possible to use a GLS procedure. One approach, which was once fairly popular, is known as the Cochrane–Orcutt procedure (see Box 5.5). Such methods work by assuming a particular form for the structure of the autocorrelation (usually a first order autoregressive process – see Chapter 6 for a general description of these models). The model would thus be specified as follows

276

(5.18) BOX 5.5 The Cochrane–Orcutt procedure (1) Assume that the general model is of the form (5.18) above. Estimate the equation in (5.18) using OLS, ignoring the residual autocorrelation (2) Obtain the residuals, and run the regression (5.19) (3) Obtain and construct etc. using this estimate of (4) Run the GLS regression (5.24). Note that a constant is not required in the specification for the errors since E(ut) = 0. If this model holds at time t, it is assumed to also hold for time t − 1, so that the model in equation (5.18) is lagged one period (5.20) Multiplying equation (5.20) by ρ (5.21) Subtracting equation (5.21) from equation (5.18) would give (5.22) Factorising, and noting that vt = ut − ρut−1 (5.23) Setting the model in equation (5.23) can be written

and

(5.24) Since the final specification equation (5.24) contains an error term that is free from autocorrelation, OLS can be directly applied to it. This 277

procedure is effectively an application of GLS. Of course, the construction of etc. requires ρ to be known. In practice, this will never be the case so that ρ has to be estimated before equation (5.24) can be used. A simple method would be to use the ρ obtained from rearranging the equation for the DW statistic given in equation (5.11). However, this is only an approximation, as the related algebra showed. This approximation may be poor in the context of small samples. The Cochrane–Orcutt procedure is an alternative, which operates as in Box 5.5. This could be the end of the process. However, Cochrane and Orcutt (1949) argue that better estimates can be obtained by going through steps (2)–(4) again. That is, given the new coefficient estimates, β2, β3, etc. construct again the residual and regress it on its previous value to obtain a new estimate for This would then be used to construct new values of the variables and a new equation (5.24) is estimated. This procedure would be repeated until the change in between one iteration and the next is less than some fixed amount (e.g., 0.01). In practice, a small number of iterations (no more than five) will usually suffice. However, the Cochrane–Orcutt procedure and similar approaches require a specific assumption to be made concerning the form of the model for the autocorrelation. Consider again equation (5.23). This can be rewritten taking ρyt−1 over to the RHS (5.25) Expanding the brackets around the explanatory variable terms would give (5.26) Now, suppose that an equation containing the same variables as (5.26) were estimated using OLS (5.27) It can be seen that equation (5.26) is a restricted version of equation (5.27), with the restrictions imposed that the coefficient on x2t in equation (5.26) multiplied by the negative of the coefficient on yt−1 gives the coefficient on x2t−1, and that the coefficient on x3t multiplied by the negative of the coefficient on yt−1 gives the coefficient on x3t−1. Thus, the restrictions 278

implied for equation (5.27) to get equation (5.26) are These are known as the common factor restrictions, and they should be tested before the Cochrane–Orcutt or similar procedure is implemented. If the restrictions hold, Cochrane–Orcutt can be validly applied. If not, however, Cochrane–Orcutt and similar techniques would be inappropriate, and the appropriate step would be to estimate an equation such as (5.27) directly using OLS. Note that in general there will be a common factor restriction for every explanatory variable (excluding a constant) x2t, x3t, …, xkt in the regression. Hendry and Mizon (1978) argued that the restrictions are likely to be invalid in practice and therefore a dynamic model that allows for the structure of y should be used rather than a residual correction on a static model – see also Hendry (1980). The White variance–covariance matrix of the coefficients (that is, calculation of the standard errors using the White correction for heteroscedasticity) is appropriate when the residuals of the estimated equation are heteroscedastic but serially uncorrelated. Newey and West (1987) develop a variance–covariance estimator that is consistent in the presence of both heteroscedasticity and autocorrelation. So an alternative approach to dealing with residual autocorrelation would be to use appropriately modified standard error estimates. While White’s correction to standard errors for heteroscedasticity as discussed above does not require any user input, the Newey–West procedure requires the specification of a truncation lag length to determine the number of lagged residuals used to evaluate the autocorrelation. The statistical software EViews, for example, uses INTEGER[4(T/100)2/9]. A more ‘modern’ view concerning autocorrelation is that it presents an opportunity rather than a problem. This view, associated with Sargan, Hendry and Mizon, suggests that serial correlation in the errors arises as a consequence of ‘misspecified dynamics’. For another explanation of the reason why this stance is taken, recall that it is possible to express the dependent variable as the sum of the parts that can be explained using the model, and a part which cannot (the residuals) (5.28) where

are

the

fitted values from the model Autocorrelation in the residuals is often 279

caused by a dynamic structure in y that has not been modelled and so has not been captured in the fitted values. In other words, there exists a richer structure in the dependent variable y and more information in the sample about that structure than has been captured by the models previously estimated. What is required is a dynamic model that allows for this extra structure in y.

5.5.8 Dynamic Models All of the models considered so far have been static in nature, e.g., (5.29) In other words, these models have allowed for only a contemporaneous relationship between the variables, so that a change in one or more of the explanatory variables at time t causes an instant change in the dependent variable at time t. But this analysis can easily be extended to the case where the current value of yt depends on previous values of y or on previous values of one or more of the variables, e.g., (5.30) It is of course possible to extend the model even more by adding further lags, e.g., x2t−2, yt−3. Models containing lags of the explanatory variables (but no lags of the explained variable) are known as distributed lag models. Specifications with lags of both explanatory and explained variables are known as autoregressive distributed lag (ADL) models. How many lags and of which variables should be included in a dynamic regression model? This is a tricky question to answer, but hopefully recourse to financial theory will help to provide an answer; for another response, see Section 5.14. Another potential ‘remedy’ for autocorrelated residuals would be to switch to a model in first differences rather than in levels. As explained previously, the first difference of yt, i.e., yt − yt−1 is denoted Δyt; similarly, one can construct a series of first differences for each of the explanatory variables, e.g., Δx2t = x2t − x2t−1, etc. Such a model has a number of other useful features (see Chapter 8 for more details) and could be expressed as (5.31) 280

Sometimes the change in y is purported to depend on previous values of the level of y or xi(i = 2, …, k) as well as changes in the explanatory variables (5.32)

5.5.9 Why Might Lags be Required in a Regression? Lagged values of the explanatory variables or of the dependent variable (or both) may capture important dynamic structure in the dependent variable that might be caused by a number of factors. Two possibilities that are relevant in finance are as follows Inertia of the dependent variable Often a change in the value of one of the explanatory variables will not affect the dependent variable immediately during one time period, but rather with a lag over several time periods. For example, the effect of a change in market microstructure or government policy may take a few months or longer to work through since agents may be initially unsure of what the implications for asset pricing are, and so on. More generally, many variables in economics and finance will change only slowly. This phenomenon arises partly as a result of pure psychological factors – for example, in financial markets, agents may not fully comprehend the effects of a particular news announcement immediately, or they may not even believe the news. The speed and extent of reaction will also depend on whether the change in the variable is expected to be permanent or transitory. Delays in response may also arise as a result of technological or institutional factors. For example, the speed of technology will limit how quickly investors’ buy or sell orders can be executed. Similarly, many investors have savings plans or other financial products where they are ‘locked in’ and therefore unable to act for a fixed period. It is also worth noting that dynamic structure is likely to be stronger and more prevalent the higher is the frequency of observation of the data. Overreactions It is sometimes argued that financial markets overreact to good and to bad news. So, for example, if a firm makes a profit warning, implying that its profits are likely to be down when formally reported later in the year, the markets might be anticipated to perceive this as implying that the value of the firm is less than was 281

previously thought, and hence that the price of its shares will fall. If there is an overreaction, the price will initially fall below that which is appropriate for the firm given this bad news, before subsequently bouncing back up to a new level (albeit lower than the initial level before the announcement). Moving from a purely static model to one which allows for lagged effects is likely to reduce, and possibly remove, serial correlation which was present in the static model’s residuals. However, other problems with the regression could cause the null hypothesis of no autocorrelation to be rejected, and these would not be remedied by adding lagged variables to the model Omission of relevant variables, which are themselves autocorrelated In other words, if there is a variable that is an important determinant of movements in y, but which has not been included in the model, and which itself is autocorrelated, this will induce the residuals from the estimated model to be serially correlated. To give a financial context in which this may arise, it is often assumed that investors assess one-step-ahead expected returns on a stock using a linear relationship (5.33) where Ωt−1 is a set of lagged information variables (i.e., Ωt−1 is a vector of observations on a set of variables at time t − 1). However, equation (5.33) cannot be estimated since the actual information set used by investors to form their expectations of returns is not known. Ωt−1 is therefore proxied with an assumed sub-set of that information, Zt−1. For example, in many popular arbitrage pricing specifications, the information set used in the estimated model includes unexpected changes in industrial production, the term structure of interest rates, inflation and default risk premia. Such a model is bound to omit some informational variables used by actual investors in forming expectations of returns, and if these are autocorrelated, it will induce the residuals of the estimated model to be also autocorrelated. Autocorrelation owing to unparameterised seasonality Suppose that the dependent variable contains a seasonal or cyclical pattern, where certain features periodically occur. This may arise, for 282

example, in the context of sales of gloves, where sales will be higher in the autumn and winter than in the spring or summer. Such phenomena are likely to lead to a positively autocorrelated residual structure that is cyclical in shape, such as that of Figure 5.4, unless the seasonal patterns are captured by the model. See Chapter 10 for a discussion of seasonality and how to deal with it. If ‘misspecification’ error has been committed by using an inappropriate functional form For example, if the relationship between y and the explanatory variables was a non-linear one, but the researcher had specified a linear regression model, this may again induce the residuals from the estimated model to be serially correlated.

5.5.10 The Long-Run Static Equilibrium Solution Once a general model of the form given in equation (5.32) has been found, it may contain many differenced and lagged terms that make it difficult to interpret from a theoretical perspective. For example, if the value of x2 were to increase in period t, what would be the effect on y in periods, t, t + 1, t + 2, and so on? One interesting property of a dynamic model that can be calculated is its long-run or static equilibrium solution. The relevant definition of ‘equilibrium’ in this context is that a system has reached equilibrium if the variables have attained some steady state values and are no longer changing, i.e., if y and x are in equilibrium, it is possible to write

Consequently, Δyt = yt − yt−1 = y − y = 0, Δx2t = x2t − x2t−1 = x2 − x2 = 0, etc. since the values of the variables are no longer changing. So the way to obtain a long-run static solution from a given empirical model such as equation (5.32) is: (1) Remove all time subscripts from the variables (2) Set error terms equal to their expected values of zero, i.e., E(ut) = 0 (3) Remove differenced terms (e.g., Δyt) altogether (4) Gather terms in x together and gather terms in y together (5) Rearrange the resulting equation if necessary so that the dependent variable y is on the LHS and is expressed as a function of the independent variables 283

EXAMPLE 5.3 Calculate the long-run equilibrium solution for the following model (5.34) Applying first steps (1)–(3) above, the static solution would be given by (5.35) Rearranging (5.35) to bring y to the LHS (5.36) and finally, dividing through by β5 (5.37) Equation (5.37) is the long-run static solution to equation (5.34). Note that this equation does not feature x3, since the only term which contained x3 was in first differenced form, so that x3 does not influence the long-run equilibrium value of y.

5.5.11 Problems with Adding Lagged Regressors to ‘Cure’ Autocorrelation In many instances, a move from a static model to a dynamic one will result in a removal of residual autocorrelation. The use of lagged variables in a regression model does, however, bring with it additional problems Inclusion of lagged values of the dependent variable violates the assumption that the explanatory variables are non-stochastic (assumption (4) of the CLRM), since by definition the value of y is determined partly by a random error term, and so its lagged values cannot be non-stochastic. In small samples, inclusion of lags of the dependent variable can lead to biased coefficient estimates, although they are still consistent, implying that the bias will disappear asymptotically (that is, as the sample size increases towards infinity). 284

What does an equation with a large number of lags actually mean? A model with many lags may have solved a statistical problem (autocorrelated residuals) at the expense of creating an interpretational one (the empirical model containing many lags or differenced terms is difficult to interpret and may not test the original financial theory that motivated the use of regression analysis in the first place). Note that if there is still autocorrelation in the residuals of a model including lags, then the OLS estimators will not even be consistent. To see why this occurs, consider the following regression model (5.38) where the errors, ut, follow a first-order autoregressive process (5.39) Substituting into equation (5.38) for ut from equation (5.39) (5.40) Now, clearly yt depends upon yt−1. Taking equation (5.38) and lagging it one period (i.e. subtracting one from each time index) (5.41) It is clear from equation (5.41) that yt−1 is related to ut−1 since they both appear in that equation. Thus, the assumption that E(X′u) = 0 is not satisfied for equation (5.41) and therefore for equation (5.38). Thus the OLS estimator will not be consistent, so that even with an infinite quantity of data, the coefficient estimates would be biased.

5.5.12 Autocorrelation in Cross-Sectional Data The possibility that autocorrelation may occur in the context of a timeseries regression is quite intuitive. However, it is also plausible that autocorrelation could be present in certain types of cross-sectional data. For example, if the cross-sectional data comprise the profitability of banks in different regions of the US, autocorrelation may arise in a spatial sense, 285

if there is a regional dimension to bank profitability that is not captured by the model. Thus the residuals from banks of the same region or in neighbouring regions may be correlated. Testing for autocorrelation in this case would be rather more complex than in the time-series context, and would involve the construction of a square, symmetric ‘spatial contiguity matrix’ or a ‘distance matrix’. Both of these matrices would be N × N, where N is the sample size. The former would be a matrix of zeros and ones, with one for element i, j when observation i occurred for a bank in the same region to, or sufficiently close to, region j and zero otherwise (i, j = 1, …, N). The distance matrix would comprise elements that measured the distance (or the inverse of the distance) between bank i and bank j. A potential solution to a finding of autocorrelated residuals in such a model would be again to use a model containing a lag structure, in this case known as a ‘spatial lag’. Further details are contained in Anselin (1988).

5.6 Assumption (4): The xt are Non-Stochastic Fortunately, it turns out that the OLS estimator is consistent and unbiased in the presence of stochastic regressors, provided that the regressors are not correlated with the error term of the estimated equation. To see this, recall that (5.42) Thus (5.43) (5.44) (5.45) Taking expectations, and provided that X and u are independent,1 (5.46) (5.47) Since E(u) = 0, this expression will be zero and therefore the estimator is still unbiased, even if the regressors are stochastic. 286

However, if one or more of the explanatory variables is contemporaneously correlated with the disturbance term, the OLS estimator will not even be consistent. This results from the estimator assigning explanatory power to the variables where in reality it is arising from the correlation between the error term and yt. Suppose for illustration that x2t and ut are positively correlated. When the disturbance term happens to take a high value, yt will also be high (because ). But if x2t is positively correlated with ut, then x2t is also likely to be high. Thus the OLS estimator will incorrectly attribute the high value of yt to a high value of x2t, where in reality yt is high simply because ut is high, which will result in biased and inconsistent parameter estimates and a fitted line that appears to capture the features of the data much better than it does in reality.

5.7 Assumption (5): The Disturbances are Normally Distributed Recall that the normality assumption (ut ~ N(0, σ2)) is required in order to conduct single or joint hypothesis tests about the model parameters.

5.7.1 Testing for Departures from Normality One of the most commonly applied tests for normality is the Bera–Jarque (hereafter BJ) test. BJ uses the property of a normally distributed random variable that the entire distribution is characterised by the first two moments – the mean and the variance. Recall from Chapter 2 that standardised third and fourth moments of a distribution are known as its skewness and kurtosis. A normal distribution is not skewed and is defined to have a coefficient of kurtosis of 3. It is possible to define a coefficient of excess kurtosis, equal to the coefficient of kurtosis minus 3; a normal distribution will thus have a coefficient of excess kurtosis of zero. Bera and Jarque (1981) formalise these ideas by testing whether the coefficient of skewness and the coefficient of excess kurtosis are jointly zero. Denoting the errors by u and their variance by σ2, it can be proved that the coefficients of skewness and kurtosis can be expressed, respectively, as (5.48)

287

The kurtosis of the normal distribution is 3 so its excess kurtosis (b2 − 3) is zero. The BJ test statistic is given by (5.49) where T is the sample size. The test statistic asymptotically follows a χ2(2) under the null hypothesis that the distribution of the series is symmetric and mesokurtic. b1 and b2 can be estimated using the residuals from the OLS regression, The null hypothesis is of normality, and this would be rejected if the residuals from the model were either significantly skewed or leptokurtic/platykurtic (or both).

5.7.2 What Should be Done if Evidence of Non-Normality is Found? It is not obvious what should be done! It is, of course, possible to employ an estimation method that does not assume normality, but such a method may be difficult to implement, and one can be less sure of its properties. It is thus desirable to stick with OLS if possible, since its behaviour in a variety of circumstances has been well researched. For sample sizes that are sufficiently large, violation of the normality assumption is virtually inconsequential. Appealing to a central limit theorem, the test statistics will asymptotically follow the appropriate distributions even in the absence of error normality.2 It is possible that a log-transform of the dependent variable might help to make the distribution of the residuals closer to a normal. This might be useful if the data are strongly positively skewed. For example, if we have cross-sectional data and we want to model company size, this is likely to be positively skewed, with the bulk of firms being of a certain size and a small number being very much larger than the rest. Alternatively, in economic or financial modelling, it is quite often the case that one or two very extreme residuals cause a rejection of the normality assumption. Such observations would appear in the tails of the distribution, and would therefore lead u4, which enters into the definition of kurtosis, to be very large. Such observations that do not fit in with the pattern of the remainder of the data are known as outliers. If this is the 288

case, one way to improve the chances of error normality is to use dummy variables or some other method to effectively remove those observations. In the time-series context, suppose that a monthly model of asset returns from 1980–90 had been estimated, and the residuals plotted, and that a particularly large outlier has been observed for October 1987, shown in Figure 5.10.

Figure 5.10 Regression residuals from stock return data, showing large

outlier for October 1987 A new variable called D87M10t could be defined as D87M10t = 1 during October 1987 and zero otherwise. The observations for the dummy variable would appear as in Box 5.6. The dummy variable would then be used just like any other variable in the regression model, e.g., (5.50) This type of dummy variable that takes the value one for only a single observation has an effect exactly equivalent to knocking out that observation from the sample altogether, by forcing the residual for that observation to zero. The estimated coefficient on the dummy variable will be equal to the residual that the dummied observation would have taken if the dummy variable had not been included. BOX 5.6 Observations for the dummy variable

289

Time

Value of dummy variable D87M10t

1986M12

0

1987M01

0

⋮

⋮

1987M09

0

1987M10

1

1987M11

0

⋮

⋮

However, many econometricians would argue that dummy variables to remove outlying residuals can be used to artificially improve the characteristics of the model – in essence fudging the results. Removing outlying observations will reduce standard errors, reduce the RSS, and therefore increase R2, thus improving the apparent fit of the model to the data. The removal of observations is also hard to reconcile with the notion in statistics that each data point represents a useful piece of information. The other side of this argument is that observations that are ‘a long way away’ from the rest, and seem not to fit in with the general pattern of the rest of the data are known as outliers. Outliers can have a serious effect on coefficient estimates since, by definition, OLS will receive a big penalty, in the form of an increased RSS for points that are a long way from the fitted line. Consequently, OLS will try extra hard to minimise the distances of points that would have otherwise been a long way from the line. A graphical depiction of the possible effect of an outlier on OLS estimation, is given in Figure 5.11.

290

Figure 5.11 Possible effect of an outlier on OLS estimation

In Figure 5.11, one point is a long way away from the rest. If this point is included in the estimation sample, the fitted line will be the dotted one, which has a slight positive slope. If this observation were removed, the full line would be the one fitted. Clearly, the slope is now large and negative. OLS would not select this line if the outlier is included since the observation is a long way from the others and hence when the residual (the distance from the point to the fitted line) is squared, it would lead to a big increase in the RSS. Note that outliers could be detected by plotting y against x only in the context of a bivariate regression. In the case where there are more explanatory variables, outliers are easiest identified by plotting the residuals over time, as in Figure 5.10, etc. So, it can be seen that a trade-off potentially exists between the need to remove outlying observations that could have an undue impact on the OLS estimates and cause residual non-normality on the one hand, and the notion that each data point represents a useful piece of information on the other. The latter is coupled with the fact that removing observations at will could artificially improve the fit of the model. A sensible way to proceed is by introducing dummy variables to the model only if there is both a statistical need to do so and a theoretical justification for their inclusion. This justification would normally come from the researcher’s knowledge of the historical events that relate to the dependent variable and the model over the relevant sample period. Dummy variables may be justifiably used to remove observations corresponding to ‘one-off’ or extreme events that are considered highly unlikely to be repeated, and the information content of which is deemed of no relevance for the data as a whole. Examples may 291

include stock market crashes, financial panics, government crises, and so on. Non-normality in financial data could also arise from certain types of heteroscedasticity, known as ARCH – see Chapter 9. In this case, the nonnormality is intrinsic to all of the data and therefore outlier removal would not make the residuals of such a model normal. Another important use of dummy variables is in the modelling of seasonality in financial data, and accounting for so-called ‘calendar anomalies’, such as day-of-the-week effects and weekend effects. These are discussed in Chapter 10.

5.8 Multicollinearity An implicit assumption that is made when using the OLS estimation method is that the explanatory variables are not correlated with one another. If there is no relationship between the explanatory variables, they would be said to be orthogonal to one another. If the explanatory variables were orthogonal to one another, adding or removing a variable from a regression equation would not cause the values of the coefficients on the other variables to change. In any practical context, the correlation between explanatory variables will be nonzero, although this will generally be relatively benign in the sense that a small degree of association between explanatory variables will almost always occur but will not cause too much loss of precision. However, a problem occurs when the explanatory variables are very highly correlated with each other, and this problem is known as multicollinearity. It is possible to distinguish between two classes of multicollinearity: perfect multicollinearity and near multicollinearity. Perfect multicollinearity occurs when there is an exact relationship between two or more variables. In this case, it is not possible to estimate all of the coefficients in the model. Perfect multicollinearity will usually be observed only when the same explanatory variable is inadvertently used twice in a regression. For illustration, suppose that two variables were employed in a regression function such that the value of one variable was always twice that of the other (e.g., suppose x3 = 2x2). If both x3 and x2 were used as explanatory variables in the same regression, then the model parameters cannot be estimated. Since the two variables are perfectly related to one another, together they contain only enough information to estimate one parameter, not two. Technically, the difficulty would occur in 292

trying to invert the (X′X) matrix since it would not be of full rank (two of the columns would be linearly dependent on one another), so that the inverse of (X′X) would not exist and hence the OLS estimates could not be calculated. Near multicollinearity is much more likely to occur in practice, and would arise when there was a non-negligible, but not perfect, relationship between two or more of the explanatory variables. Note that a high correlation between the dependent variable and one of the independent variables is not multicollinearity. Visually, we could think of the difference between near and perfect multicollinearity as follows. Suppose that the variables x2t and x3t were highly correlated. If we produced a scatter plot of x2t against x3t, then perfect multicollinearity would correspond to all of the points lying exactly on a straight line, while near multicollinearity would correspond to the points lying close to the line, and the closer they were to the line (taken altogether), the stronger would be the relationship between the two variables.

5.8.1 Measuring Near Multicollinearity Testing for multicollinearity is surprisingly difficult, and hence all that is presented here is a simple method to investigate the presence or otherwise of the most easily detected forms of near multicollinearity. This method simply involves looking at the matrix of correlations between the individual variables. Suppose that a regression equation has three explanatory variables (plus a constant term), and that the pairwise correlations between these explanatory variables are

Clearly, if multicollinearity was suspected, the most likely culprit would be a high correlation between x2 and x4. Of course, if the relationship involves three or more variables that are collinear – e.g., x2 + x3 ≈ x4 – then multicollinearity would be very difficult to detect. A more formal method for measuring the extent of multicollinearity is by calculating the variance inflation factors (VIF), which provide an 293

estimate of to what extent the variance of a parameter estimate increases because the explanatory variables are correlated. For example, if a VIF for a particular variable is 4, this suggests that the variance of the parameter estimate is 4 times larger than would be the case if it were independent from the other explanatory variables in the model (so that the standard error is twice as large (i.e., the square root of 4). The VIF can be calculated for variable i as (5.51) where is the R2 value from an auxiliary regression of the explanatory variable i on an intercept plus all of the other explanatory variables from the model. The larger the VIF, the more serious is the collinearity between the explanatory variable under test and the others in the model. It is clear from equation (5.51) that since (under some assumptions), the R2 will be positive, the minimum value of VIF is one, and this would take place if the variable under study is independent of all of the other explanatory variables. As a rule of thumb, usually if the VIF is below 5, multicollinearity is usually assumed to be negligible, whereas if it is greater than or equal to 5, the problem is sufficiently serious that some remedial action is warranted. Some researchers use a threshold of 10, rather than 5, to indicate whether multicollinearity is sufficiently large to be cause for concern.

5.8.2 Problems if Near Multicollinearity is Present but Ignored First, R2 will be high but the individual coefficients will have high standard errors, so that the regression ‘looks good’ as a whole, but the individual variables are not significant.3 This arises in the context of very closely related explanatory variables as a consequence of the difficulty in observing the individual contribution of each variable to the overall fit of the regression. Second, the regression becomes very sensitive to small changes in the specification, so that adding or removing an explanatory variable leads to large changes in the coefficient values or significances of the other variables. Finally, near multicollinearity will thus make confidence intervals for the parameters very wide, and significance tests might therefore give inappropriate conclusions, and so make it difficult to draw sharp inferences. 294

5.8.3 Solutions to the Problem of Multicollinearity A number of alternative estimation techniques have been proposed that are valid in the presence of multicollinearity – for example, ridge regression, or principal components. Principal components analysis (PCA) was discussed briefly in Appendix 4.1 to Chapter 4. Many researchers do not use these techniques, however, as they can be complex, their properties are less well understood than those of the OLS estimator and, above all, many econometricians would argue that multicollinearity is more a problem with the data than with the model or estimation method. Other, more ad hoc methods for dealing with the possible existence of near multicollinearity include Ignore it, if the model is otherwise adequate, i.e., statistically and in terms of each coefficient being of a plausible magnitude and having an appropriate sign. Sometimes, the existence of multicollinearity does not reduce the t-ratios on variables that would have been significant without the multicollinearity sufficiently to make them insignificant. It is worth stating that the presence of near multicollinearity does not affect the BLUE properties of the OLS estimator – i.e., it will still be consistent, unbiased and efficient since the presence of near multicollinearity does not violate any of the CLRM assumptions (1)–(4). However, in the presence of near multicollinearity, it will be hard to obtain small standard errors. This will not matter if the aim of the model-building exercise is to produce forecasts from the estimated model, since the forecasts will be unaffected by the presence of near multicollinearity so long as this relationship between the explanatory variables continues to hold over the forecasted sample. Drop one of the collinear variables, so that the problem disappears. However, this may be unacceptable to the researcher if there were strong a priori theoretical reasons for including both variables in the model. Also, if the removed variable was relevant in the data generating process for y, an omitted variable bias would result (see Section 5.10). Transform the highly correlated variables into a ratio and include only the ratio and not the individual variables in the regression. Again, this may be unacceptable if financial theory suggests that changes in the dependent variable should occur following changes in the individual explanatory variables, and not a ratio of them. Finally, as stated above, it is also often said that near multicollinearity 295

is more a problem with the data than with the model, so that there is insufficient information in the sample to obtain estimates for all of the coefficients. This is why near multicollinearity leads coefficient estimates to have wide standard errors, which is exactly what would happen if the sample size were small. An increase in the sample size will usually lead to an increase in the accuracy of coefficient estimation and consequently a reduction in the coefficient standard errors, thus enabling the model to better dissect the effects of the various explanatory variables on the explained variable. A further possibility, therefore, is for the researcher to go out and collect more data – for example, by taking a longer run of data, or switching to a higher frequency of sampling. Of course, it may be infeasible to increase the sample size if all available data are being utilised already. A further method of increasing the available quantity of data as a potential remedy for near multicollinearity would be to use a pooled sample. This would involve the use of data with both cross-sectional and time-series dimensions (see Chapter 11).

5.9 Adopting the Wrong Functional Form A further implicit assumption of the classical linear regression model is that the appropriate ‘functional form’ is linear. This means that the appropriate model is assumed to be linear in the parameters, and that in the bivariate case, the relationship between y and x can be represented by a straight line. However, this assumption may not always be upheld. Whether the model should be linear can be formally tested using Ramsey’s (1969) RESET test, which is a general test for misspecification of functional form. Essentially, the method works by using higher order terms of the fitted values (e.g., etc.) in an auxiliary regression. The auxiliary regression is thus one where yt, the dependent variable from the original regression, is regressed on powers of the fitted values together with the original explanatory variables (5.52) Higher order powers of the fitted values of y can capture a variety of nonlinear relationships, since they embody higher order powers and crossproducts of the original explanatory variables, e.g., (5.53) 296

We are interested in testing the joint null hypothesis that α2 = 0 and α3 = 0 and … and αp = 0. Note that in some applications, equation (5.52) is broken into two stages: first a standard linear regression is undertaken and the residuals (call these following the conventional notation) are collected. Then the become the dependent variable in a second stage regression that includes the powers of the fitted values etc., with only the latter and a constant included in the auxiliary regression (5.54) The residuals in the auxiliary regression, would be the same in both 2 cases. The value of R is obtained from the regression (5.52), and the test statistic, given by TR2, is distributed asymptotically as a χ2(p − 1). Note that the degrees of freedom for this test will be (p − 1) and not p. This arises because p is the highest order term in the fitted values used in the auxiliary regression and thus the test will involve p − 1 terms, one for the square of the fitted value, one for the cube, …, one for the pth power. If the value of the test statistic is greater than the χ2 critical value, reject the null hypothesis that the functional form was correct.

5.9.1 What if the Functional Form is Found to be Inappropriate? One possibility would be to switch to a non-linear model, but the RESET test presents the user with no guide as to what a better specification might be! Also, non-linear models in the parameters typically preclude the use of OLS, and require the use of a non-linear estimation technique. Some nonlinear models can still be estimated using OLS, provided that they are linear in the parameters. For example, if the true model is of the form (5.55) – that is, a second order polynomial (i.e., a quadratic) in x – and the researcher assumes that the relationship between yt and xt is linear (so is missing from the specification), this is simply a special case of omitted variables, with the usual problems (see Section 5.10) and obvious remedy. Sometimes a quadratic form of equation such as that in equation (5.55) is useful to allow for a relationship where y increases with x2 at an increasing rate, or where y initially increases as x increases but then the 297

increase tails off and eventually reverses as x increases further. These two situations are displayed in Figure 5.12. In the left-hand graph, y increases at an increasing rate (at least over the relevant range of values of x) and both β2 and β3 would be positive. By contrast, the right-hand figure shows the situation where β2 is positive but β3 is negative. As we saw in Chapter 3, in this situation we would obtain a ∩=shape for the curve since the squared term would come to dominate the overall behaviour of the function as x increases. To offer an example of where this might be relevant, it has been suggested in the research literature that the relationship between age and attitude to risk is non-linear, so that risk tolerance rises with age for a certain range (e.g., from 18–40 years) and then declines thereafter – we could capture this with a quadratic where x is years of age and y is some measure or risk tolerance.

Figure 5.12 Relationship between y and x2 in a quadratic regression for

different values of β2 and β3 We could add even higher order terms to equation (5.55), such as a cubic or quartic term. It might be that a cubic would be useful to capture something like a point of inflection, where the relationship between x and y hits a stationary point but it is rare that we would be able to justify any higher order term than a quadratic from the perspective of its relevance. An alternative possibility is that the model may be multiplicatively nonlinear or have a more complex relationship that cannot be captured by simply adding higher order powers of explanatory variables to the 298

regression model. Therefore, another approach that is sensible in this case would be to transform the data into logarithms. This will linearise many previously multiplicative models into additive ones. For example, consider again the exponential growth model (5.56) Taking logs, this becomes (5.57) or (5.58) where Yt = ln(yt), α = ln(β1), Xt = ln(xt), vt = ln(ut). Thus a simple logarithmic transformation makes this model a standard linear bivariate regression equation that can be estimated using OLS. Loosely following the treatment given in Stock and Watson (2011), the following list shows four different functional forms for models that are either linear or can be made linear following a logarithmic transformation to one or more of the dependent or independent variables, examining only a bivariate specification for simplicity. Care is needed when interpreting the coefficient values in each case. (1) Linear model: yt = β1 + β2x2t + ut; a 1-unit increase in x2t causes a β2unit increase in yt.

(2) Log-linear: ln(yt) = β1 + β2x2t + ut; a 1-unit increase in x2t causes a 100 × β2% increase in yt.

299

(3) Linear-log: yt = β1 + β2ln(x2t) + ut; a 1% increase in x2t causes a 0.01 × β2- unit increase in yt.

(4) Double log: ln(yt) = β1 + β2ln(x2t) + ut; a 1% increase in x2t causes a β2% increase in yt. Note that to plot y against x2 would be more complex since the shape would depend on the size of β2.

Note also that we cannot use R2 or adjusted R2 to determine which of these four types of model is most appropriate since the dependent variables are different across some of the models.

5.10 Omission of an Important Variable What would be the effects of excluding from the estimated regression a 300

variable that is a determinant of the dependent variable? For example, suppose that the true, but unknown, data generating process is represented by (5.59) but the researcher estimated a model of the form (5.60) so that the variable x5t is omitted from the model. The consequence would be that the estimated coefficients on all the other variables will be biased and inconsistent unless the excluded variable is uncorrelated with all the included variables. Even if this condition is satisfied, the estimate of the coefficient on the constant term will be biased, which would imply that any forecasts made from the model would be biased. The standard errors will also be biased (upwards), and hence hypothesis tests could yield inappropriate inferences. Further intuition is offered in Dougherty (1992, pp. 168–73).

5.11 Inclusion of an Irrelevant Variable Suppose now that the researcher makes the opposite error to Section 5.10, i.e., that the true data generating process (DGP) was represented by (5.61) but the researcher estimates a model of the form (5.62) thus incorporating the superfluous or irrelevant variable x5t. As x5t is irrelevant, the expected value of β5 is zero, although in any practical application, its estimated value is very unlikely to be exactly zero. The consequence of including an irrelevant variable would be that the coefficient estimators would still be consistent and unbiased, but the estimators would be inefficient. This would imply that the standard errors for the coefficients are likely to be inflated relative to the values which they would have taken if the irrelevant variable had not been included. Variables which would otherwise have been marginally significant may no 301

longer be so in the presence of irrelevant variables. In general, it can also be stated that the extent of the loss of efficiency will depend positively on the absolute value of the correlation between the included irrelevant variable and the other explanatory variables. Summarising the last two sections it is evident that when trying to determine whether to err on the side of including too many or too few variables in a regression model, there is an implicit trade-off between inconsistency and efficiency; many researchers would argue that while in an ideal world, the model will incorporate precisely the correct variables – no more and no less – the former problem is more serious than the latter and therefore in the real world, one should err on the side of incorporating marginally significant variables.

5.12 Parameter Stability Tests So far, regressions of a form such as (5.63) have been estimated. These regressions embody the implicit assumption that the parameters (β1, β2 and β3) are constant for the entire sample, both for the data period used to estimate the model, and for any subsequent period used in the construction of forecasts. This implicit assumption can be tested using parameter stability tests. The idea is essentially to split the data into sub-periods and then to estimate up to three models, for each of the sub-parts and for all the data and then to ‘compare’ the RSS of each of the models. There are two types of test that will be considered, namely the Chow (analysis of variance) test and predictive failure tests.

5.12.1 The Chow Test The steps involved are shown in Box 5.7. Note that it is also possible to use a dummy variables approach to calculating both Chow and predictive failure tests. In the case of the Chow test, the unrestricted regression would contain dummy variables for the intercept and for all of the slope coefficients (see also Chapter 10). For example, suppose that the regression is of the form (5.64) 302

If the split of the total of T observations is made so that the sub-samples contain T1 and T2 observations (where T1 + T2 = T), the unrestricted regression would be given by (5.65) where Dt = 1 for t ∈ T1 and zero otherwise. In other words, Dt takes the value one for observations in the first sub-sample and zero for observations in the second subsample. The Chow test viewed in this way would then be a standard F-test of the joint restriction H0: β4 = 0 and β5 = 0 and β6 = 0, with equations (5.64) and (5.65) being the unrestricted and restricted regressions, respectively. BOX 5.7 Conducting a Chow test (1) Split the data into two sub-periods Estimate the regression over the whole period and then for the two sub-periods separately (three regressions). Obtain the RSS for each regression. (2) The restricted regression is now the regression for the whole period while the ‘unrestricted regression’ comes in two parts: one for each of the sub-samples. It is thus possible to form an Ftest, which is based on the difference between the RSSs. The statistic is (5.66) where RSS = residual sum of squares for whole sample RSS1 = residual sum of squares for sub-sample 1 RSS2 = residual sum of squares for sub-sample 2 T = number of observations 2k = number of regressors in the ‘unrestricted’ regression (since it comes in two parts) k = number of regressors in (each) ‘unrestricted’ regression The unrestricted regression is the one where the restriction has not been imposed on the model. Since the restriction is that the coefficients are equal across the sub-samples, the restricted regression will be the single regression for the whole sample. 303

Thus, the test is one of how much the residual sum of squares for the whole sample (RSS) is bigger than the sum of the residual sums of squares for the two sub-samples (RSS1 + RSS2). If the coefficients do not change much between the samples, the residual sum of squares will not rise much upon imposing the restriction. Thus the test statistic in equation (5.66) can be considered a straightforward application of the standard F-test formula discussed in Chapter 4. The restricted residual sum of squares in equation (5.66) is RSS, while the unrestricted residual sum of squares is (RSS1 + RSS2). The number of restrictions is equal to the number of coefficients that are estimated for each of the regressions, i.e., k. The number of regressors in the unrestricted regression (including the constants) is 2k, since the unrestricted regression comes in two parts, each with k regressors. (3) Perform the test If the value of the test statistic is greater than the critical value from the F-distribution, which is an F(k, T − 2k), then reject the null hypothesis that the parameters are stable over time.

EXAMPLE 5.4 Suppose that it is now January 1993. Consider the following regression for the standard CAPM β for the returns on a stock (5.67) where rgt and rMt are excess returns on Glaxo shares and on a market portfolio, respectively. Suppose that you are interested in estimating beta using monthly data from 1981 to 1992, to aid a stock selection decision. Another researcher expresses concern that the October 1987 stock market crash fundamentally altered the risk–return relationship. Test this conjecture using a Chow test. The model for each sub-period is 1981M1−1987M10 (5.68)

304

1987M11–1992M12 (5.69) 1981M1–1992M12 (5.70) The null hypothesis is where the subscripts 1 and 2 denote the parameters for the first and second sub-samples, respectively. The test statistic will be given by (5.71) The test statistic should be compared with a 5%, F(2, 140) = 3.06. H0 is rejected at the 5% level and hence it is concluded that the restriction that the coefficients are the same in the two periods cannot be employed. The appropriate modelling response would probably be to employ only the second part of the data in estimating the CAPM beta relevant for investment decisions made in early 1993.

5.12.2 The Predictive Failure Test A problem with the Chow test is that it is necessary to have enough data to do the regression on both sub-samples, i.e., T1 ≫ k, T2 ≫ k. This may not hold in the situation where the total number of observations available is small. Even more likely is the situation where the researcher would like to examine the effect of splitting the sample at some point very close to the start or very close to the end of the sample. An alternative formulation of a test for the stability of the model is the predictive failure test, which requires estimation for the full sample and one of the sub-samples only. The predictive failure test works by estimating the regression over a ‘long’ sub-period (i.e., most of the data) and then using those coefficient estimates for predicting values of y for the other period. These predictions for y are then implicitly compared with the actual values. Although it can be expressed in several different ways, the null hypothesis for this test is 305

that the prediction errors for all of the forecasted observations are zero. To calculate the test: Run the regression for the whole period (the restricted regression) and obtain the RSS. Run the regression for the ‘large’ sub-period and obtain the RSS (called RSS1). Note that in this book, the number of observations for the long estimation sub-period will be denoted by T1 (even though it may come second). The test statistic is given by (5.72) where T2 = number of observations that the model is attempting to ‘predict’. The test statistic will follow an F(T2, T1 − k). For an intuitive interpretation of the predictive failure test statistic formulation, consider an alternative way to test for predictive failure using a regression containing dummy variables. A separate dummy variable would be used for each observation that was in the prediction sample. The unrestricted regression would then be the one that includes the dummy variables, which will be estimated using all T observations, and will have (k + T2) regressors (the k original explanatory variables, and a dummy variable for each prediction observation, i.e., a total of T2 dummy variables). Thus the numerator of the last part of equation (5.72) would be the total number of observations (T) minus the number of regressors in the unrestricted regression (k + T2). Noting also that T − (k + T2) = (T1 − k), since T1 + T2 = T, this gives the numerator of the last term in equation (5.72). The restricted regression would then be the original regression containing the explanatoryvariables but none of the dummy variables. Thus the number of restrictions would be the number of observations in the prediction period, which would be equivalent to the number of dummy variables included in the unrestricted regression, T2. To offer an illustration, suppose that the regression is again of the form of (5.64), and that the last three observations in the sample are used for a predictive failure test. The unrestricted regression would include three dummy variables, one for each of the observations in T2

306

(5.73) where D1t = 1 for observation T− 2 and zero otherwise, D2t = 1 for observation T − 1 and zero otherwise, D3t = 1 for observation T and zero otherwise. In this case, k = 2, and T2 = 3. The null hypothesis for the predictive failure test in this regression is that the coefficients on all of the dummy variables are zero (i.e., H0 : γ1 = 0 and γ2 = 0 and γ3 = 0). Both approaches to conducting the predictive failure test described above are equivalent, although the dummy variable regression is likely to take more time to set up. However, for both the Chow and the predictive failure tests, the dummy variables approach has the one major advantage that it provides the user with more information. This additional information comes from the fact that one can examine the significances of the coefficients on the individual dummy variables to see which part of the joint null hypothesis is causing a rejection. For example, in the context of the Chow regression, is it the intercept or the slope coefficients that are significantly different across the two sub-samples? In the context of the predictive failure test, use of the dummy variables approach would show for which period(s) the prediction errors are significantly different from zero.

5.12.3 Backward versus Forward Predictive Failure Tests There are two types of predictive failure tests – forward tests and backwards tests. Forward predictive failure tests are where the last few observations are kept back for forecast testing. For example, suppose that observations for 1980Q1–2013Q4 are available. A forward predictive failure test could involve estimating the model over 1980Q1–2012Q4 and forecasting 2013Q1–2013Q4. Backward predictive failure tests attempt to ‘back-cast’ the first few observations, e.g., if data for 1980Q1–2013Q4 are available, and the model is estimated over 1981Q1–2013Q4 and back-cast 1980Q1–1980Q4. Both types of test offer further evidence on the stability of the regression relationship over the whole sample period. EXAMPLE 5.5 Suppose that the researcher decided to determine the stability of the estimated model for stock returns over the whole sample in Example 5.4 by using a predictive failure test of the last two years of observations. The following models would be estimated 307

1981M1–1992M12 (whole sample) (5.74) 1981M1–1990M12 (‘long sub-sample’) (5.75) Can this regression adequately ‘forecast’ the values for the last two years? The test statistic would be given by (5.76) Compare the test statistic with an F(24, 118) = 1.66 at the 5% level. So the null hypothesis that the model can adequately predict the last few observations would not be rejected. It would thus be concluded that the model did not suffer from predictive failure during the 1991M1– 1992M12 period.

5.12.4 How Can the Appropriate Sub-Parts to Use be Decided? As a rule of thumb, some or all of the following methods for selecting where the overall sample split occurs could be used: Plot the dependent variable over time and split the data accordingly to any obvious structural changes in the series, as illustrated in Figure 5.13. It is clear that y underwent a large fall in its value around observation 175, and it is possible that this may have caused a change in its behaviour. A Chow test could be conducted with the sample split at this observation. Split the data according to any known important historical events (e.g., a stock market crash, change in market microstructure, new government elected). The argument is that a major change in the underlying environment in which y is measured is more likely to cause a structural change in the model’s parameters than a relatively trivial change. Use all but the last few observations and do a forward predictive failure test on those. Use all but the first few observations and do a backward predictive 308

failure test on those.

Figure 5.13 Plot of a variable showing suggestion for break date

If a model is good, it will survive a Chow or predictive failure test with any break date. If the Chow or predictive failure tests are failed, two approaches could be adopted. Either the model is respecified, for example, by including additional variables, or separate estimations are conducted for each of the sub-samples. On the other hand, if the Chow and predictive failure tests show no rejections, it is empirically valid to pool all of the data together in a single regression. This will increase the sample size and therefore the number of degrees of freedom relative to the case where the sub-samples are used in isolation.

5.12.5 The QLR Test The Chow and predictive failure tests will work satisfactorily if the date of a structural break in a financial time series can be specified. But more often, a researcher will not know the break date in advance, or may know only that it lies within a given range (sub-set) of the sample period. In such circumstances, a modified version of the Chow test, known as the Quandt likelihood ratio (QLR) test, named after Quandt (1960), can be used instead. The test works by automatically computing the usual Chow F-test statistic repeatedly with different break dates, then the break date giving the largest F-statistic value is chosen. While the test statistic is of the Fvariety, it will follow a non-standard distribution rather than an Fdistribution since we are selecting the largest from a number of F-statistics rather than examining a single one. 309

The test is well behaved only when the range of possible break dates is sufficiently far from the end points of the whole sample, so it is usual to ‘trim’ the sample by (typically) 5% at each end. To illustrate, suppose that the full sample comprises 200 observations; then we would test for a structural break between observations 31 and 170 inclusive. The critical values will depend on how much of the sample is trimmed away, the number of restrictions under the null hypothesis (the number of regressors in the original regression as this is effectively a Chow test), and the significance level.

5.12.6 Stability Tests Based on Recursive Estimation An alternative to the QLR test for use in the situation where a researcher believes that a series may contain a structural break but is unsure of the date is to perform a recursive estimation. This is sometimes known as recursive least squares (RLS). The procedure is appropriate only for timeseries data or cross-sectional data that have been ordered in some sensible way (for example, a sample of annual stock returns, ordered by market capitalisation). Recursive estimation simply involves starting with a subsample of the data, estimating the regression, then sequentially adding one observation at a time and rerunning the regression until the end of the sample is reached. It is common to begin the initial estimation with the very minimum number of observations possible, which will be k + 1. So at the first step, the model is estimated using observations 1 to k + 1; at the second step, observations 1 to k + 2 are used and so on; at the final step, observations 1 to T are used. The final result will be the production of T − k separate estimates of every parameter in the regression model. It is to be expected that the parameter estimates produced near the start of the recursive procedure will appear rather unstable since these estimates are being produced using so few observations, but the key question is whether they then gradually settle down or whether the volatility continues through the whole sample. Seeing the latter would be an indication of parameter instability. It should be evident that RLS in itself is not a statistical test for parameter stability as such, but rather it provides qualitative information which can be plotted and thus gives a very visual impression of how stable the parameters appear to be. But two important stability tests, known as the CUSUM and CUSUMSQ tests, are derived from the residuals of the recursive estimation (known as the recursive residuals).4 The CUSUM statistic is based on a normalised (i.e., scaled) version of the cumulative 310

sums of the residuals. Under the null hypothesis of perfect parameter stability, the CUSUM statistic is zero however many residuals are included in the sum (because the expected value of a disturbance is always zero). A set of ±2 standard error bands is usually plotted around zero and any statistic lying outside the bands is taken as evidence of parameter instability. The CUSUMSQ test is based on a normalised version of the cumulative sums of squared residuals. The scaling is such that under the null hypothesis of parameter stability, the CUSUMSQ statistic will start at zero and end the sample with a value of 1. Again, a set of ±2 standard error bands is usually plotted around zero and any statistic lying outside these is taken as evidence of parameter instability.

5.13 Measurement Errors As stated above, one of the of the assumptions of the classical linear regression model is that the explanatory variables are non-stochastic. One way in which this assumption can be violated is when there is a two-way causal relationship between the explanatory and explained variable, and this situation (simultaneous equations bias) is discussed in detail in Chapter 7. A further situation where the assumption will not apply is when there is measurement error in one or more of the explanatory variables. Sometimes this is also known as the errors-in-variables problem. Measurement errors can occur in a variety of circumstances – for example, macroeconomic variables are almost always estimated quantities (GDP, inflation and so on), as is most information contained in company accounts. Similarly, it is sometimes the case that we cannot observe or obtain data on a variable we require and so we need to use a proxy variable – for instance, many models include expected quantities (e.g., expected inflation) but since we cannot typically measure expectations, we need to use a proxy. More generally, measurement error could be present in the dependent or independent variables, and each of these cases is considered in the following sub-sections.

5.13.1 Measurement Error in the Explanatory Variable(s) For simplicity, suppose that we wish to estimate a model containing just one explanatory variable, xt (5.77) 311

where ut is a disturbance term. Suppose further that xt is measured with error so that instead of observing its true value, we observe a noisy version, that comprises the actual xt plus some additional noise, vt, that is independent of xt and ut (5.78) Taking equation (5.77) and substituting in for xt from equation (5.78), we get (5.79) We can rewrite this equation by separately expressing the composite error term, (ut − β2vt) (5.80) It should be clear from equations (5.78) and (5.80) that the explanatory variable measured with error, and the composite error term (ut − β2vt) are correlated since both depend on vt. Thus the requirement that the explanatory variables are non-stochastic does not hold. This causes the parameters to be estimated inconsistently. It can be shown that the size of the bias in the estimates will be a function of the variance of the noise in xt as a proportion of the overall disturbance variance. It can be further shown that if β2 is positive, the bias will be negative but if β2 is negative, the bias will be positive – in other words, the parameter estimate will always be biased towards zero as a result of the measurement noise. The impact of this estimation bias when the explanatory variables are measured with error can be quite important and is a serious issue in particular when testing asset pricing models. The standard approach to testing the CAPM pioneered by Fama and MacBeth (1973) comprises two stages (discussed more fully in Chapter 14). Stage one is to run separate time-series regressions for each firm to estimate the betas and the second stage involves running a cross-sectional regression of the stock returns on the betas. Since the betas are estimated at the first stage rather than being directly observable, they will surely contain measurement error. In the finance literature, the effect of this has sometimes been termed attenuation bias. Early tests of the CAPM showed that the relationship between beta and returns was positive but smaller than expected, and this is precisely 312

what would happen as a result of measurement error in the betas. Various approaches to solving this issue have been proposed, the most common of which is to use portfolio betas in place of individual stock betas in the second stage. The hope is that this will smooth out the estimation error in the betas. An alternative approach attributed to Shanken (1992) is to modify the standard errors in the second-stage regression to adjust directly for the measurement errors in the betas. More discussion of this issue will be presented in Chapter 15.

5.13.2 Measurement Error in the Explained Variable Measurement error in the explained variable is much less serious than in the explanatory variable(s); recall that one of the motivations for the inclusion of the disturbance term in a regression model is that it can capture measurement errors in y. Thus, when the explained variable is measured with error, the disturbance term will in effect be a composite of the usual disturbance term and another source of noise from the measurement error. In such circumstances, the parameter estimates will still be consistent and unbiased and the usual formulae for calculating standard errors will still be appropriate. The only consequence is that the additional noise means that the standard errors will be enlarged relative to the situation where there was no measurement error in y.

5.14 A Strategy for Constructing Econometric Models and a Discussion of Model-Building Philosophies The objective of many econometric model-building exercises is to build a statistically adequate empirical model which satisfies the assumptions of the CLRM, is parsimonious, has the appropriate theoretical interpretation, and has the right ‘shape’ (i.e., all signs on coefficients are ‘correct’ and all sizes of coefficients are ‘correct’). But how might a researcher go about achieving this objective? A common approach to model building is the ‘LSE’ or general-to-specific methodology associated with Sargan and Hendry. This approach essentially involves starting with a large model which is statistically adequate and restricting and rearranging the model to arrive at a parsimonious final formulation. Hendry’s approach (see Gilbert, 1986) argues that a good model is consistent with the data and with theory. A good model will also encompass rival models, which means that it can explain all that rival models can and more. The Hendry methodology 313

suggests the extensive use of diagnostic tests to ensure the statistical adequacy of the model. An alternative philosophy of econometric model-building, which predates Hendry’s research, is that of starting with the simplest model and adding to it sequentially so that it gradually becomes more complex and a better description of reality. This approach, associated principally with Koopmans (1937), is sometimes known as a ‘specific-to-general’ or ‘bottoms-up’ modelling approach. Gilbert (1986) termed this the ‘Average Economic Regression’ since most applied econometric work had been tackled in that way. This term was also having a joke at the expense of a top economics journal that published many papers using such a methodology. Hendry and his co-workers have severely criticised this approach, mainly on the grounds that diagnostic testing is undertaken, if at all, almost as an after-thought and in a very limited fashion. However, if diagnostic tests are not performed, or are performed only at the end of the modelbuilding process, all earlier inferences are potentially invalidated. Moreover, if the specific initial model is generally misspecified, the diagnostic tests themselves are not necessarily reliable in indicating the source of the problem. For example, if the initially specified model omits relevant variables which are themselves autocorrelated, introducing lags of the included variables would not be an appropriate remedy for a significant DW test statistic. Thus the eventually selected model under a specific-togeneral approach could be sub-optimal in the sense that the model selected using a general-to-specific approach might represent the data better. Under the Hendry approach, diagnostic tests of the statistical adequacy of the model come first, with an examination of inferences for financial theory drawn from the model left until after a statistically adequate model has been found. According to Hendry and Richard (1982), a final acceptable model should satisfy several criteria (adapted slightly here). The model should: be logically plausible be consistent with underlying financial theory, including satisfying any relevant parameter restrictions have regressors that are uncorrelated with the error term have parameter estimates that are stable over the entire sample have residuals that are white noise (i.e., completely random and exhibiting no patterns) be capable of explaining the results of all competing models and more 314

The last of these is known as the encompassing principle. A model that nests within it a smaller model always trivially encompasses it. But a small model is particularly favoured if it can explain all of the results of a larger model; this is known as parsimonious encompassing. The advantages of the general-to-specific approach are that it is statistically sensible and also that the theory on which the models are based usually has nothing to say about the lag structure of a model. Therefore, the lag structure incorporated in the final model is largely determined by the data themselves. Furthermore, the statistical consequences from excluding relevant variables are usually considered more serious than those from including irrelevant variables. The general-to-specific methodology is conducted as follows. The first step is to form a ‘large’ model with lots of variables on the RHS. This is known as a generalised unrestricted model (GUM), which should originate from financial theory, and which should contain all variables thought to influence the dependent variable. At this stage, the researcher is required to ensure that the model satisfies all of the assumptions of the CLRM. If the assumptions are violated, appropriate actions should be taken to address or allow for this, e.g., taking logs, adding lags, adding dummy variables. It is important that the steps above are conducted prior to any hypothesis testing. It should also be noted that the diagnostic tests presented above should be cautiously interpreted as general rather than specific tests. In other words, rejection of a particular diagnostic test null hypothesis should be interpreted as showing that there is something wrong with the model. So, for example, if the RESET test or White’s test show a rejection of the null, such results should not be immediately interpreted as implying that the appropriate response is to find a solution for inappropriate functional form or heteroscedastic residuals, respectively. It is quite often the case that one problem with the model could cause several assumptions to be violated simultaneously. For example, an omitted variable could cause failures of the RESET, heteroscedasticity and autocorrelation tests. Equally, a small number of large outliers could cause non-normality and residual autocorrelation (if they occur close together in the sample) and heteroscedasticity (if the outliers occur for a narrow range of the explanatory variables). Moreover, the diagnostic tests themselves do not operate optimally in the presence of other types of misspecification since they essentially assume that the model is correctly specified in all other respects. For example, it is not clear that tests for heteroscedasticity will behave well if the residuals are autocorrelated. 315

Once a model that satisfies the assumptions of the CLRM has been obtained, it could be very big, with large numbers of lags and independent variables. The next stage is therefore to reparameterise the model by knocking out very insignificant regressors. Also, some coefficients may be insignificantly different from each other, so that they can be combined. At each stage, it should be checked whether the assumptions of the CLRM are still upheld. If this is the case, the researcher should have arrived at a statistically adequate empirical model that can be used for testing underlying financial theories, forecasting future values of the dependent variable, or for formulating policies. However, needless to say, the general-to-specific approach also has its critics. For small or moderate sample sizes, it may be impractical. In such instances, the large number of explanatory variables will imply a small number of degrees of freedom. This could mean that none of the variables is significant, especially if they are highly correlated. This being the case, it would not be clear which of the original long list of candidate regressors should subsequently be dropped. Moreover, in any case the decision on which variables to drop may have profound implications for the final specification of the model. A variable whose coefficient was not significant might have become significant at a later stage if other variables had been dropped instead. In theory, sensitivity of the final specification to the various possible paths of variable deletion should be carefully checked. However, this could imply checking many (perhaps even hundreds) of possible specifications. It could also lead to several final models, none of which appears noticeably better than the others. The general-to-specific approach, if followed faithfully to the end, will hopefully lead to a statistically valid model that passes all of the usual model diagnostic tests and contains only statistically significant regressors. However, the final model could also be a bizarre creature that is devoid of any theoretical interpretation. There would also be more than just a passing chance that such a model could be the product of a statistically vindicated data mining exercise. Such a model would closely fit the sample of data at hand, but could fail miserably when applied to other samples if it is not based soundly on theory. There now follows another example of the use of the classical linear regression model in finance, based on an examination of the determinants of sovereign credit ratings by Cantor and Packer (1996).

316

5.15 Determinants of Sovereign Credit Ratings 5.15.1 Background Sovereign credit ratings are an assessment of the riskiness of debt issued by governments. They embody an estimate of the probability that the borrower will default on her obligation. Two famous US ratings agencies, Moody’s and Standard and Poor’s (S&P), provide ratings for many governments. Although the two agencies use different symbols to denote the given riskiness of a particular borrower, the ratings of the two agencies are comparable. Gradings are split into two broad categories: investment grade and speculative grade. Investment grade issuers have good or adequate payment capacity, while speculative grade issuers either have a high degree of uncertainty about whether they will make their payments, or are already in default. The highest grade offered by the agencies, for the highest quality of payment capacity, is ‘triple A’, which Moody’s denotes ‘Aaa’ and S&P denotes ‘AAA’. The lowest grade issued to a sovereign in the Cantor and Packer sample was B3 (Moody’s) or B–(S&P). Thus the number of grades of debt quality from the highest to the lowest given to governments in their sample is 16. The central aim of Cantor and Packer’s paper is an attempt to explain and model how the agencies arrived at their ratings. Although the ratings themselves are publicly available, the models or methods used to arrive at them are shrouded in secrecy. The agencies also provide virtually no explanation as to what the relative weights of the factors that make up the rating are. Thus, a model of the determinants of sovereign credit ratings could be useful in assessing whether the ratings agencies appear to have acted rationally. Such a model could also be employed to try to predict the rating that would be awarded to a sovereign that has not previously been rated and when a re-rating is likely to occur. The paper continues, among other things, to consider whether ratings add to publicly available information, and whether it is possible to determine what factors affect how the sovereign yields react to ratings announcements.

5.15.2 Data Cantor and Packer (1996) obtain a sample of government debt ratings for forty-nine countries as of September 1995 that range between the above gradings. The ratings variable is quantified, so that the highest credit quality (Aaa/AAA) in the sample is given a score of 16, while the lowest rated sovereign in the sample is given a score of 1 (B3/B–). This score 317

forms the dependent variable. The factors that are used to explain the variability in the ratings scores are macroeconomic variables. All of these variables embody factors that are likely to influence a government’s ability and willingness to service its debt costs. Ideally, the model would also include proxies for socio-political factors, but these are difficult to measure objectively and so are not included. It is not clear in the paper from where the list of factors was drawn. The included variables (with their units of measurement) are: Per capita income (in 1994 US dollars, thousands). Cantor and Packer argue that per capita income determines the tax base, which in turn influences the government’s ability to raise revenue. GDP growth (annual 1991–4 average, %). The growth rate of increase in GDP is argued to measure how much easier it will become to service debt costs in the future. Inflation (annual 1992–4 average, %). Cantor and Packer argue that high inflation suggests that inflationary money financing will be used to service debt when the government is unwilling or unable to raise the required revenue through the tax system. Fiscal balance (average annual government budget surplus as a proportion of GDP 1992–4, %). Again, a large fiscal deficit shows that the government has a relatively weak capacity to raise additional revenue and to service debt costs. External balance (average annual current account surplus as a proportion of GDP 1992–4, %). Cantor and Packer argue that a persistent current account deficit leads to increasing foreign indebtedness, which may be unsustainable in the long run. External debt (foreign currency debt as a proportion of exports in 1994, %). Reasoning as for external balance (which is the change in external debt over time). Dummy for economic development (=1 for a country classified by the International Monetary Fund (IMF) as developed, 0 otherwise). Cantor and Packer argue that credit ratings agencies perceive developing countries as relatively more risky beyond that suggested by the values of the other factors listed above. Dummy for default history (=1 if a country has defaulted, 0 otherwise). It is argued that countries that have previously defaulted experience a large fall in their credit rating. The income and inflation variables are transformed to their logarithms. 318

The model is linear and estimated using OLS. Some readers of this book who have a background in econometrics will note that strictly, OLS is not an appropriate technique when the dependent variable can take on only one of a certain limited set of values (in this case, 1, 2, 3, …16). In such applications, a technique such as ordered probit (not covered in this text) would usually be more appropriate. Cantor and Packer argue that any approach other than OLS is infeasible given the relatively small sample size (forty-nine), and the large number (sixteen) of ratings categories. The results from regressing the rating value on the variables listed above are presented in their exhibit 5, adapted and presented here as Table 5.2. Four regressions are conducted, each with identical independent variables but a different dependent variable. Regressions are conducted for the rating score given by each agency separately, with results presented in columns (4) and (5) of Table 5.2. Occasionally, the ratings agencies give different scores to a country – for example, in the case of Italy, Moody’s gives a rating of ‘A1’, which would generate a score of 12 on a 16-scale. S&P, on the other hand, gives a rating of ‘AA’, which would score 14 on the 16-scale, two gradings higher. Thus a regression with the average score across the two agencies, and with the difference between the two scores as dependent variables, is also conducted, and presented in columns (3) and (6), respectively, of Table 5.2. Table 5.2 Determinants and impacts of sovereign credit ratings Dependent variable Explanatory variable (1)

Expected sign (2)

Average rating (3)

Moody’s rating (4)

S&P rating (5)

Difference Moody’s/S&P (6)

Intercept

?

1.442 (0.663)

3.408 (1.379)

−0.524 (−0.223)

3.932 (2.521)

Per capita income

+

1.242*** (5.302)

1.027*** (4.041)

1.458*** (6.048)

−0.431 (−2.688)

GDP growth

+

0.151 (1.935)

0.130 (1.545)

0.171** (2.132)

−0.040 (0.756)

Inflation

−

−0.611*** (−2.839)

−0.630*** (−2.701)

−0.591*** (−2.671)

−0.039 (−0.265)

Fiscal

+

0.073

0.049

0.097*

−0.048

319

balance

(1.324)

(0.818)

(1.71)

(−1.274)

External balance

+

0.003 (0.314)

0.006 (0.535)

0.001 (0.046)

0.006 (0.779)

External debt

−

−0.013*** (−5.088)

−0.015*** (−5.365)

−0.011*** (−4.236)

−0.004 (−2.133)

Development dummy

+

2.776*** (4.25)

2.957*** (4.175)

2.595*** (3.861)

0.362 (0.81)

Default dummy

−

−2.042*** (−3.175)

−1.63** (−2.097)

−2.622*** (−3.962)

1.159 (2.632)

0.924

0.905

0.926

0.836

Adjusted R2

Notes: t-ratios in parentheses; *, ** and *** indicate significance at the 10%, 5% and 1% levels, respectively. Source: Cantor and Packer (1996). Reprinted with permission from Institutional Investor.

5.15.3 Interpreting the Models The models are difficult to interpret in terms of their statistical adequacy, since virtually no diagnostic tests have been undertaken. The values of the adjusted R2, at over 90% for each of the three ratings regressions, are high for cross-sectional regressions, indicating that the model seems able to capture almost all of the variability of the ratings about their mean values across the sample. There does not appear to be any attempt at reparameterisation presented in the paper, so it is assumed that the authors reached this set of models after some searching. In this particular application, the residuals have an interesting interpretation as the difference between the actual and fitted ratings. The actual ratings will be integers from 1 to 16, although the fitted values from the regression and therefore the residuals can take on any real value. Cantor and Packer argue that the model is working well as no residual is bigger than 3, so that no fitted rating is more than three categories out from the actual rating, and only four countries have residuals bigger than two categories. Furthermore, 70% of the countries have ratings predicted exactly (i.e., the residuals are less than 0.5 in absolute value). Now, turning to interpret the models from a financial perspective, it is of interest to investigate whether the coefficients have their expected signs and sizes. The expected signs for the regression results of columns (3)–(5) 320

are displayed in column (2) of Table 5.2 (as determined by this author). As can be seen, all of the coefficients have their expected signs, although the fiscal balance and external balance variables are not significant or are only very marginally significant in all three cases. The coefficients can be interpreted as the average change in the rating score that would result from a unit change in the variable. So, for example, a rise in per capita income of $1,000 will on average increase the rating by 1.0 units according to Moody’s and 1.5 units according to S&P. The development dummy suggests that, on average, a developed country will have a rating three notches higher than an otherwise identical developing country. And everything else equal, a country that has defaulted in the past will have a rating two notches lower than one that has always kept its obligation. By and large, the ratings agencies appear to place similar weights on each of the variables, as evidenced by the similar coefficients and significances across columns (4) and (5) of Table 5.2. This is formally tested in column (6) of the table, where the dependent variable is the difference between Moody’s and S&P ratings. Only three variables are statistically significantly differently weighted by the two agencies. S&P places higher weights on income and default history, while Moody’s places more emphasis on external debt.

5.15.4 The Relationship Between Ratings and Yields In this section of the paper, Cantor and Packer try to determine whether ratings have any additional information useful for modelling the crosssectional variability of sovereign yield spreads over and above that contained in publicly available macroeconomic data. The dependent variable is now the log of the yield spread, i.e., ln(Yield on the sovereign bond – Yield on a US Treasury Bond) One may argue that such a measure of the spread is imprecise, for the true credit spread should be defined by the entire credit quality curve rather than by just two points on it. However, leaving this issue aside, the results are presented in Table 5.3. Table 5.3 Do ratings add to public information?

Variable

Expected sign

Dependent variable: ln (yield spread) (1) 321

(2)

(3)

Intercept

?

2.105*** (16.148)

Average rating

–

–0.221*** (–19.175)

Per capita income

–

–0.144 (–0.927)

0.226 (1.523)

GDP growth

–

–0.004 (–0.142)

0.029 (1.227)

Inflation

+

0.108 (1.393)

–0.004 (–0.068)

Fiscal balance

–

–0.037 (–1.557)

–0.02 (–1.045)

External balance

–

–0.038 (–1.29)

–0.023 (–1.008)

External debt

+

0.003*** (2.651)

0.000 (0.095)

Development dummy

–

–0.723*** (–2.059)

–0.38 (–1.341)

Default dummy

+

0.612*** (2.577)

0.085 (0.385)

0.857

0.914

Adjusted R2

0.919

0.466 (0.345)

0.074 (0.071) –0.218*** (–4.276)

Notes: t-ratios in parentheses; *, **and *** indicate significance at the 10%, 5% and 1% levels, respectively. Source: Cantor and Packer (1996). Reprinted with permission from Institutional Investor.

Three regressions are presented in Table 5.3, denoted specifications (1), (2) and (3). The first of these is a regression of the ln(spread) on only a constant and the average rating (column (1)), and this shows that ratings have a highly significant inverse impact on the spread. Specification (2) is a regression of the ln(spread) on the macroeconomic variables used in the previous analysis. The expected signs are given (as determined by this author) in column (2). As can be seen, all coefficients have their expected signs, although now only the coefficients belonging to the external debt and the two dummy variables are statistically significant. Specification (3) 322

is a regression on both the average rating and the macroeconomic variables. When the rating is included with the macroeconomic factors, none of the latter is any longer significant – only the rating coefficient is statistically significantly different from zero. This message is also portrayed by the adjusted R2 values, which are highest for the regression containing only the rating, and slightly lower for the regression containing the macroeconomic variables and the rating. One may also observe that, under specification (3), the coefficients on the per capita income, GDP growth and inflation variables now have the wrong sign. This is, in fact, never really an issue, for if a coefficient is not statistically significant, it is indistinguishable from zero in the context of hypothesis testing, and therefore it does not matter whether it is actually insignificant and positive or insignificant and negative. Only coefficients that are both of the wrong sign and statistically significant imply that there is a problem with the regression. It would thus be concluded from this part of the paper that there is no more incremental information in the publicly available macroeconomic variables that is useful for predicting the yield spread than that embodied in the rating. The information contained in the ratings encompasses that contained in the macroeconomic variables.

5.15.5 What Determines How the Market Reacts to Ratings Announcements? Cantor and Packer also consider whether it is possible to build a model to predict how the market will react to ratings announcements, in terms of the resulting change in the yield spread. The dependent variable for this set of regressions is now the change in the log of the relative spread, i.e., log[(yield – treasury yield)/treasury yield], over a two-day period at the time of the announcement. The sample employed for estimation comprises every announcement of a ratings change that occurred between 1987 and 1994; seventy-nine such announcements were made, spread over eighteen countries. Of these, thirty nine were actual ratings changes by one or more of the agencies, and forty were listed as likely in the near future to experience a regrading. Moody’s calls this a ‘watchlist’, while S&P term it their ‘outlook’ list. The explanatory variables are mainly dummy variables for whether the announcement was positive – i.e., an upgrade whether there was an actual ratings change or just listing for probable 323

regrading whether the announcement was made by Moody’s or S&P whether the bond was speculative grade or investment grade whether there had been another ratings announcement in the previous sixty days the ratings gap between the announcing and the other agency The following cardinal variable was also employed: the change in the spread over the previous sixty days The results are presented in Table 5.4, but in this text, only the final specification (numbered 5 in Cantor and Packer’s exhibit 11) containing all of the variables described above is included. Table 5.4 What determines reactions to ratings announcements? Dependent variable: log relative spread Independent variable

Coefficient (u–ratio)

Intercept

–0.02 (–1.4)

Positive announcements

0.01 (0.34)

Ratings changes

–0.01 (–0.37)

Moody’s announcements

0.02 (1.51)

Speculative grade

0.03** (2.33)

Change in relative spreads from day –60 to day –1

–0.06 (–1.1)

Rating gap

0.03* (1.7)

Other rating announcements from day – 60 to day –1

0.05** (2.15)

324

Adjusted R2

0.12

Note: * and ** denote significance at the 10% and 5% levels, respectively. Source: Cantor and Packer (1996). Reprinted with permission from Institutional Investor.

As can be seen from Table 5.4, the models appear to do a relatively poor job of explaining how the market will react to ratings announcements. The adjusted R2 value is only 12%, and this is the highest of the five specifications tested by the authors. Further, only two variables are significant and one marginally significant of the seven employed in the model. It can therefore be stated that yield changes are significantly higher following a ratings announcement for speculative than investment grade bonds, and that ratings changes have a bigger impact on yield spreads if there is an agreement between the ratings agencies at the time the announcement is made. Further, yields change significantly more if there has been a previous announcement in the past sixty days than if not. On the other hand, neither whether the announcement is an upgrade or a downgrade, nor whether it is an actual ratings change or a name on the watchlist, nor whether the announcement is made by Moody’s or S&P, nor the amount by which the relative spread has already changed over the past sixty days, has any significant impact on how the market reacts to ratings announcements.

5.15.6 Conclusions To summarise, six factors appear to play a big role in determining sovereign credit ratings – incomes, GDP growth, inflation, external debt, industrialised or not and default history The ratings provide more information on yields than all of the macroeconomic factors put together One cannot determine with any degree of confidence what factors determine how the markets will react to ratings announcements. KEY CONCEPTS The key terms to be able to define and explain from this chapter are homoscedasticity autocorrelation equilibrium solution 325

skewness outlier multicollinearity irrelevant variable recursive least squares measurement error heteroscedasticity dynamic model robust standard errors kurtosis functional form omitted variable parameter stability general-to-specific approach

SELF-STUDY QUESTIONS 1. Are assumptions made concerning the unobservable error terms (ut) or about their sample counterparts, the estimated residuals Explain your answer. 2. What pattern(s) would one like to see in a residual plot and why? 3. A researcher estimates the following model for stock market returns, but thinks that there may be a problem with it. By calculating the t-ratios and considering their significance and by examining the value of R2 or otherwise, suggest what the problem might be. (5.81) How might you go about solving the perceived problem? 4. (a) State in algebraic notation and explain the assumption about the CLRM’s disturbances that is referred to by the term ‘homoscedasticity’. (b) What would the consequence be for a regression model if the errors were not homoscedastic? (c) How might you proceed if you found that (b) were actually 326

5. (a) (b)

(c)

(d)

the case? What do you understand by the term ‘autocorrelation’? An econometrician suspects that the residuals of her model might be autocorrelated. Explain the steps involved in testing this theory using the Durbin–Watson (DW) test. The econometrician follows your guidance (!!!) in part (b) and calculates a value for the Durbin–Watson statistic of 0.95. The regression has sixty quarterly observations and three explanatory variables (plus a constant term). Perform the test. What is your conclusion? In order to allow for autocorrelation, the econometrician decides to use a model in first differences with a constant (5.82)

By attempting to calculate the long-run solution to this model, explain what might be a problem with estimating models entirely in first differences. (e) The econometrician finally settles on a model with both first differences and lagged levels terms of the variables (5.83) Can the Durbin–Watson test still validly be used in this case? 6. Calculate the long-run static equilibrium solution to the following dynamic econometric model (5.84) 7. What might Ramsey’s RESET test be used for? What could be done if it were found that the RESET test has been failed? 8. (a) Why is it necessary to assume that the disturbances of a regression model are normally distributed? (b) In a practical econometric modelling situation, how might the problem that the residuals are not normally distributed be addressed? 327

9. (a) Explain the term ‘parameter structural stability’? (b) A financial econometrician thinks that the stock market crash of October 1987 fundamentally changed the risk–return relationship given by the CAPM equation. He decides to test this hypothesis using a Chow test. The model is estimated using monthly data from January 1981–December 1995, and then two separate regressions are run for the sub-periods corresponding to data before and after the crash. The model is (5.85) so that the excess return on a security at time t is regressed upon the excess return on a proxy for the market portfolio at time t. The results for the three models estimated for a given stock are as follows: 1981M1–1995M12 (5.86) 1981M1–1987M10 (5.87) 1987M11–1995M12 (5.88) (c) What are the null and alternative hypotheses that are being tested here, in terms of α and β? (d) Perform the test. What is your conclusion? 10. For the same model as above, and given the following results, do a forward and backward predictive failure test: 1981M1–1995M12 (5.89) 1981M1–1994M12

328

(5.90) 1982M1–1995M12 (5.91) What is your conclusion? 11. Why is it desirable to remove insignificant variables from a regression? 12. Explain why it is not possible to include an outlier dummy variable in a regression model when you are conducting a Chow test for parameter stability. Will the same problem arise if you were to conduct a predictive failure test? Why or why not? 13. (a) Explain the term ‘measurement error’. (b) How does measurement error arise? (c) Is measurement error more serious if it is present in the dependent variable or the independent variable(s) of a regression? Explain your answer. (d) What is the likely impact of measurement error on tests of the CAPM and what are the possible solutions?

1 2

3 4

A situation where X and u are not independent is discussed at length in Chapter 7. The law of large numbers states that the average of a sample (which is a random variable) will converge to the population mean (which is fixed), and the central limit theorem states that the sample mean converges to a normal distribution. Note that multicollinearity does not affect the value of R2 in a regression. Strictly, the CUSUM and CUSUMSQ statistics are based on the one-stepahead prediction errors – i.e., the differences between yt and its predicted value based on the parameters estimated at time t − 1. See Greene (2002, Chapter 7) for full technical details.

329

6 Univariate Time-Series Modelling and Forecasting

LEARNING OUTCOMES In this chapter, you will learn how to Explain the defining characteristics of various types of stochastic processes Identify the appropriate time-series model for a given data series Produce forecasts for autoregressive moving average (ARMA) and exponential smoothing models Evaluate the accuracy of predictions using various metrics

6.1 Introduction Univariate time-series models are a class of specifications where one attempts to model and to predict financial variables using only information contained in their own past values and possibly current and past values of an error term. This practice can be contrasted with structural models, which are multivariate in nature, and attempt to explain changes in a variable by reference to the movements in the current or past values of other (explanatory) variables. Time series models are usually a-theoretical, implying that their construction and use is not based upon any underlying theoretical model of the behaviour of a variable. Instead, time series models are an attempt to capture empirically relevant features of the observed data that may have arisen from a variety of different (but unspecified) structural models. An important class of time series models is 330

the family of autoregressive integrated moving average (ARIMA) models, usually associated with Box and Jenkins (1976). Time series models may be useful when a structural model is inappropriate. For example, suppose that there is some variable yt whose movements a researcher wishes to explain. It may be that the variables thought to drive movements of yt are not observable or not measurable, or that these forcing variables are measured at a lower frequency of observation than yt. For instance, yt might be a series of daily stock returns, where possible explanatory variables could be macroeconomic indicators that are available monthly. Additionally, as will be examined later in this chapter, structural models are often not useful for out-of-sample forecasting. These observations motivate the consideration of pure time series models, which are the focus of this chapter. The approach adopted for this topic is as follows. In order to define, estimate and use ARIMA models, one first needs to specify the notation and to define several important concepts. The chapter will then consider the properties and characteristics of a number of specific models from the ARIMA family. The book endeavours to answer the following question: ‘For a specified time series model with given parameter values, what will be its defining characteristics?’ Following this, the problem will be reversed, so that the reverse question is asked: ‘Given a set of data, with characteristics that have been determined, what is a plausible model to describe that data?’

6.2 Some Notation and Concepts The following sub-sections define and describe several important concepts in time-series analysis. Each will be elucidated and drawn upon later in the chapter. The first of these concepts is the notion of whether a series is stationary or not. Determining whether a series is stationary or not is very important, for the stationarity or otherwise of a series can strongly influence its behaviour and properties. Further detailed discussion of stationarity, testing for it, and implications of it not being present, are covered in Chapter 8.

6.2.1 A Strictly Stationary Process A strictly stationary process is one where, for any t1, t2, …, tT ∈ Z, any k ∈ Z and T = 1,2, … 331

(6.1) where F denotes the joint distribution function of the set of random variables (Tong, 1990, p.3). It can also be stated that the probability measure for the sequence {yt} is the same as that for {yt+k}∀ k (where ‘∀ k’ means ‘for all values of k’). In other words, a series is strictly stationary if the distribution of its values remains the same as time progresses, implying that the probability that y falls within a particular interval is the same now as at any time in the past or the future.

6.2.2 A Weakly Stationary Process If a series satisfies (6.2)–(6.4) for t = 1, 2, …, ∞, it is said to be weakly or covariance stationary (6.2) (6.3) (6.4) These three equations state that a stationary process should have a constant mean, a constant variance and a constant autocovariance structure, respectively. Definitions of the mean and variance of a random variable are probably well known to readers, but the autocovariances may not be. The autocovariances determine how y is related to its previous values, and for a stationary series they depend only on the difference between t1 and t2, so that the covariance between yt and yt−1 is the same as the covariance between yt−10 and yt−11, etc. The moment (6.5) is known as the autocovariance function. When s = 0, the autocovariance at lag zero is obtained, which is the autocovariance of yt with yt, i.e., the variance of y. These covariances, γs, are also known as autocovariances since they are the covariances of y with its own previous values. The autocovariances are not a particularly useful measure of the relationship between y and its previous values, however, since the values of the autocovariances depend on the units of measurement of yt, and hence the 332

values that they take have no immediate interpretation. It is thus more convenient to use the autocorrelations, which are the autocovariances normalised by dividing by the variance (6.6) The series τs now has the standard property of correlation coefficients that the values are bounded to lie between ±1. In the case that s = 0, the autocorrelation at lag zero is obtained, i.e., the correlation of yt with yt, which is of course 1. If τs is plotted against s = 0, 1, 2, …, a graph known as the autocorrelation function (acf) or correlogram is obtained.

6.2.3 A White Noise Process Roughly speaking, a white noise process is one with no discernible structure. A definition of a white noise process is (6.7) (6.8) (6.9) Thus a white noise process has constant mean and variance, and zero autocovariances, except at lag zero. Another way to state this last condition would be to say that each observation is uncorrelated with all other values in the sequence. Hence the autocorrelation function for a white noise process will be zero apart from a single peak of 1 at s = 0. If μ = 0, and the three conditions hold, the process is known as zero mean white noise. If it is further assumed that yt is distributed normally, then the sample autocorrelation coefficients are also approximately normally distributed where T is the sample size, and denotes the autocorrelation coefficient at lag s estimated from a sample. This result can be used to conduct significance tests for the autocorrelation coefficients by constructing a non-rejection region (like a confidence interval) for an estimated autocorrelation coefficient to determine whether it is significantly different 333

from zero. For example, a 95% non-rejection region would be given by

for s ≠ 0. If the sample autocorrelation coefficient, falls outside this region for a given value of s, then the null hypothesis that the true value of the coefficient at that lag s is zero is rejected. It is also possible to test the joint hypothesis that all m of the τk correlation coefficients are simultaneously equal to zero using the Qstatistic developed by Box and Pierce (1970) (6.10) where T = sample size, m = maximum lag length. The correlation coefficients are squared so that the positive and negative coefficients do not cancel each other out. Since the sum of squares of independent standard normal variates is itself a χ2 variate with degrees of freedom equal to the number of squares in the sum, it can be stated that the Q-statistic is asymptotically distributed as a under the null hypothesis that all m autocorrelation coefficients are zero. As for any joint hypothesis test, only one autocorrelation coefficient needs to be statistically significant for the test to result in a rejection. However, the Box–Pierce test has poor small sample properties, implying that it leads to the wrong decision too frequently for small samples. A variant of the Box–Pierce test, having better small sample properties, has been developed. The modified statistic is known as the Ljung–Box (1978) statistic (6.11) It should be clear from the form of the statistic that asymptotically (that is, as the sample size increases towards infinity), the (T + 2) and (T − k) terms in the Ljung–Box formulation will cancel out, so that the statistic is equivalent to the Box–Pierce test. This statistic is very useful as a portmanteau (general) test of linear dependence in time series. EXAMPLE 6.1 334

Suppose that a researcher had estimated the first five autocorrelation coefficients using a series of length 100 observations, and found them to be Lag Autocorrelation coefficient

1 0.207

2 −0.013

3 0.086

4 0.005

5 −0.022

Test each of the individual correlation coefficients for significance, and test all five jointly using the Box–Pierce and Ljung–Box tests. SOLUTION A 95% confidence interval can be constructed for each coefficient using

where T = 100 in this case. The decision rule is thus to reject the null hypothesis that a given coefficient is zero in the cases where the coefficient lies outside the range (−0.196, +0.196). For this example, it would be concluded that only the first autocorrelation coefficient is significantly different from zero at the 5% level. Now, turning to the joint tests, the null hypothesis is that all of the first five autocorrelation coefficients are jointly zero, i.e. The test statistics for the Box–Pierce and Ljung–Box tests are given respectively, as (6.12)

(6.13)

The relevant critical values are from a χ2 distribution with five degrees of freedom, which are 11.1 at the 5% level, and 15.1 at the 1% level. Clearly, 335

in both cases, the joint null hypothesis that all of the first five autocorrelation coefficients are zero cannot be rejected. Note that, in this instance, the individual test caused a rejection while the joint test did not. This is an unexpected result that may have arisen as a result of the low power of the joint test when four of the five individual autocorrelation coefficients are insignificant. Thus the effect of the significant autocorrelation coefficient is diluted in the joint test by the insignificant coefficients. The sample size used in this example is also modest relative to those commonly available in finance.

6.3 Moving Average Processes The simplest class of time-series model that one could entertain is that of the moving average process. Let ut (t = 1, 2, 3, …) be a white noise process with E(ut) = 0 and var(ut) = σ2. Then (6.14) is a qth order moving average model, denoted MA(q). This can be expressed using sigma notation as (6.15) A moving average model is simply a linear combination of white noise processes, so that yt depends on the current and previous values of a white noise disturbance term. Equation equation (6.15) will later have to be manipulated, and such a process is most easily achieved by introducing the lag operator notation. This would be written Lyt = yt−1 to denote that yt is lagged once. In order to show that the ith lag of yt is being taken (that is, the value that yt took i periods ago), the notation would be Liyt = yt−i. Note that in some books and studies, the lag operator is referred to as the ‘backshift operator’, denoted by B. Using the lag operator notation, (6.15) would be written as (6.16) or as 336

(6.17) where: In much of what follows, the constant (μ) is dropped from the equations. Removing μ considerably eases the complexity of algebra involved, and is inconsequential for it can be achieved without loss of generality. To see this, consider a sample of observations on a series, zt that has a mean A zero-mean series, yt can be constructed by simply subtracting from each observation zt. The distinguishing properties of the moving average process of order q given above are (1)

(6.18)

(2)

(6.19)

(3) (6.20)

So, a moving average process has constant mean, constant variance, and autocovariances which may be non-zero to lag q and will always be zero thereafter. Each of these results will be derived below. EXAMPLE 6.2 Consider the following MA(2) process (6.21) where ut is a zero mean white noise process with variance σ2. (1) Calculate the mean and variance of yt. (2) Derive the autocorrelation function for this process (i.e., express the autocorrelations, τ1, τ2, … as functions of the parameters θ1 and θ2). (3) If θ1 = −0.5 and θ2 = 0.25, sketch the acf of yt. 337

SOLUTION (1)

(6.22) So the expected value of the error term is zero for all time periods. Taking expectations of both sides of equation (6.21) gives (6.23) (6.24) but E(yt) = 0, so that the last component in each set of square brackets in equation (6.24) is zero and this reduces to (6.25) Replacing yt in equation (6.25) with the RHS of (6.21) (6.26) (6.27) But E[cross-products] = 0 since cov(ut, ut−s) = 0 for s ≠ 0. ‘Crossproducts’ is thus a catchall expression for all of the terms in u which have different time subscripts, such as ut−1 ut−2 or ut−5 ut−20, etc. Again, one does not need to worry about these cross-product terms, since these are effectively the autocovariances of ut, which will all be zero by definition since ut is a random error process, which will have zero autocovariances (except at lag zero). So (6.28) (6.29) (6.30) γ0 can also be interpreted as the autocovariance at lag zero.

(2) Calculating now the acf of yt, first determine the autocovariances and then the autocorrelations by dividing the autocovariances by the 338

variance. The autocovariance at lag 1 is given by (6.31) (6.32) (6.33) Again, ignoring the cross-products, equation (6.33) can be written as (6.34) (6.35) (6.36) The autocovariance at lag 2 is given by (6.37) (6.38) (6.39) (6.40) (6.41) The autocovariance at lag 3 is given by (6.42) (6.43) (6.44) (6.45) So γs = 0 for s > 2. All autocovariances for the MA(2) process will be zero for any lag length, s, greater than 2. The autocorrelation at lag 0 is given by (6.46) The autocorrelation at lag 1 is given by (6.47) 339

The autocorrelation at lag 2 is given by (6.48) The autocorrelation at lag 3 is given by (6.49) The autocorrelation at lag s is given by (6.50) (3) For θ1 = −0.5 and θ2 = 0.25, substituting these into the formulae above gives the first two autocorrelation coefficients as τ1 = −0.476, τ2 = 0.190. Autocorrelation coefficients for lags greater than 2 will all be zero for an MA(2) model. Thus the acf plot will appear as in Figure 6.1.

Figure 6.1 Autocorrelation function for sample MA(2) process

6.4 Autoregressive Processes An autoregressive model is one where the current value of a variable, y, 340

depends upon only the values that the variable took in previous periods plus an error term. An autoregressive model of order p, denoted as AR(p), can be expressed as (6.51) where ut is a white noise disturbance term. A manipulation of expression (13.24) will be required to demonstrate the properties of an autoregressive model. This expression can be written more compactly using sigma notation (6.52) or using the lag operator, as (6.53) or (6.54) where

6.4.1 The Stationarity Condition Stationarity is a desirable property of an estimated AR model, for several reasons. One important reason is that a model whose coefficients are nonstationary will exhibit the unfortunate property that previous values of the error term will have a non-declining effect on the current value of yt as time progresses. This is arguably counter-intuitive and empirically implausible in many cases. More discussion on this issue will be presented in Chapter 8. Box 6.1 defines the stationarity condition algebraically. BOX 6.1 The stationarity condition for an AR(p) model Setting μ to zero in equation (6.54), for a zero mean AR(p) process, yt, given by

341

(6.55) it would be stated that the process is stationary if it is possible to write (6.56) with ϕ (L)−1 converging to zero. This means that the autocorrelations will decline eventually as the lag length is increased. When the expansion ϕ (L)−1 is calculated, it will contain an infinite number of terms, and can be written as an MA(∞), e.g., If the process given by equation (6.54) is stationary, the coefficients in the MA(∞) representation will decline eventually with lag length. On the other hand, if the process is nonstationary, the coefficients in the MA(∞) representation would not converge to zero as the lag length increases. The condition for testing for the stationarity of a general AR(p) model is that the roots of the ‘characteristic equation’ (6.57) all lie outside the unit circle. The notion of a characteristic equation is so-called because its roots determine the characteristics of the process yt – for example, the acf for an AR process will depend on the roots of this characteristic equation, which is a polynomial in z. EXAMPLE 6.3 Is the following model stationary? (6.58) In order to test this, first write yt−1 in lag operator notation (i.e., as Lyt), and take this term over to the LHS of equation (6.58), and factorise (6.59) (6.60) (6.61) Then the characteristic equation is 342

(6.62) having the root z = 1, which lies on, not outside, the unit circle. In fact, the particular AR(p) model given by equation (6.58) is a non-stationary process known as a random walk (see Chapter 8). This procedure can also be adopted for autoregressive models with longer lag lengths and where the stationarity or otherwise of the process is less obvious. For example, is the following process for yt stationary? (6.63) Again, the first stage is to express this equation using the lag operator notation, and then taking all the terms in y over to the LHS (6.64) (6.65) The characteristic equation is (6.66) which fortunately factorises to (6.67) so that the roots are z = 1, z = 2/3, and z = 2. Only one of these lies outside the unit circle and hence the process for yt described by equation (6.63) is not stationary.

6.4.2 Wold’s Decomposition Theorem Wold’s decomposition theorem states that any stationary series can be decomposed into the sum of two unrelated processes, a purely deterministic part and a purely stochastic part, which will be an MA(∞). A simpler way of stating this in the context of AR modelling is that any stationary autoregressive process of order p with no constant and no other terms can be expressed as an infinite order moving average model. This result is important for deriving the autocorrelation function for an autoregressive process. 343

For the AR(p) model, given in, for example, equation (6.51) (with μ set to zero for simplicity) and expressed using the lag polynomial notation, ϕ (L)yt = ut, the Wold decomposition is (6.68) where The characteristics of an autoregressive process are as follows. The (unconditional) mean of y is given by (6.69) The autocovariances and autocorrelation functions can be obtained by solving a set of simultaneous equations known as the Yule–Walker equations. The Yule–Walker equations express the correlogram (the τ s) as a function of the autoregressive coefficients (the ϕs)

(6.70)

For any AR model that is stationary, the autocorrelation function will decay geometrically to zero.1 These characteristics of an autoregressive process will be derived from first principles below using an illustrative example. EXAMPLE 6.4 Consider the following simple AR(1) model (6.71) (1) Calculate the (unconditional) mean of yt. For the remainder of the question, set the constant to zero (μ = 0) for simplicity. (2) Calculate the (unconditional) variance of yt. (3) Derive the autocorrelation function for this process. 344

SOLUTION (i) The unconditional mean will be given by the expected value of expression (6.71) (6.72) (6.73) But also (6.74) So, replacing yt−1 in (6.73) with the RHS of (6.72) (6.75) (6.76) Lagging equation (6.74) by a further one period (6.77) Repeating the steps given above one more time (6.78) (6.79) Hopefully, readers will by now be able to see a pattern emerging. Making n such substitutions would give (6.80) So long as the model is stationary, i.e., |ϕ1| < 1, then Therefore, taking limits as n → ∞, then and so (6.81) Recall the rule of algebra that the finite sum of an infinite number of geometrically declining terms in a series is given by ‘first term in 345

series divided by (1 minus common difference)’, where the common difference is the quantity that each term in the series is multiplied by to arrive at the next term. It can thus be stated from (6.81) that (6.82) Thus the expected or mean value of an autoregressive process of order one is given by the intercept parameter divided by one minus the autoregressive coefficient. (ii) Calculating now the variance of yt, with μ set to zero (6.83) This can be written equivalently as (6.84) From Wold’s decomposition theorem, the AR(p) can be expressed as an MA(∞) (6.85) (6.86) or (6.87) So long as |ϕ1| < 1, i.e., so long as the process for yt is stationary, this sum will converge. From the definition of the variance of any random variable y, it is possible to write (6.88) but E(yt) = 0, since μ is set to zero to obtain equation (6.83) above. Thus (6.89) (6.90) 346

(6.91) As discussed above, the ‘cross-products’ can be set to zero. (6.92) (6.93) (6.94) Provided that |ϕ1| < 1, the infinite sum in equation (6.94) can be written as (6.95) (iii) Turning now to the calculation of the autocorrelation function, the auto-covariances must first be calculated. This is achieved by following similar algebraic manipulations as for the variance above, starting with the definition of the autocovariances for a random variable. The autocovariances for lags 1, 2, 3, …, s, will be denoted by γ1, γ2, γ3, …, γs, as previously. (6.96) Since μ has been set to zero, E(yt) = 0 and E(yt−1) = 0, so (6.97) under the result above that E(yt) = E(yt−1) = 0. Thus (6.98) (6.99) Again, the cross-products can be ignored so that (6.100) (6.101) (6.102)

347

For the second autocovariance, (6.103) Using the same rules as applied above for the lag 1 covariance (6.104) (6.105) (6.106) (6.107) (6.108) (6.109) By now it should be possible to see a pattern emerging. If these steps were repeated for γ3, the following expression would be obtained (6.110) and for any lag s, the autocovariance would be given by (6.111) The acf can now be obtained by dividing the covariances by the variance, so that (6.112)

(6.113)

348

(6.114)

(6.115) The autocorrelation at lag s is given by (6.116) which means that Note that use of the Yule–Walker equations would have given the same answer.

6.5 The Partial Autocorrelation Function The partial autocorrelation function, or pacf (denoted τkk), measures the correlation between an observation k periods ago and the current observation, after controlling for observations at intermediate lags (i.e., all lags < k) – i.e. the correlation between yt and yt−k, after removing the effects of yt−k+1, yt−k+2, …, yt−1. For example, the pacf for lag 3 would measure the correlation between yt and yt−3 after controlling for the effects of yt−1 and yt−2. At lag 1, the autocorrelation and partial autocorrelation coefficients are equal, since there are no intermediate lag effects to eliminate. Thus, τ11 = τ1, where τ1 is the autocorrelation coefficient at lag 1. At lag 2 (6.117) where τ1 and τ2 are the autocorrelation coefficients at lags 1 and 2, respectively. For lags greater than two, the formulae are more complex and hence a presentation of these is beyond the scope of this book. There now proceeds, however, an intuitive explanation of the characteristic shape of the pacf for a moving average and for an autoregressive process. In the case of an autoregressive process of order p, there will be direct connections between yt and yt−s for s ≤ p, but no direct connections for s > p. For example, consider the following AR(3) model 349

(6.118) There is a direct connection through the model between yt and yt−1, and between yt and yt−2, and between yt and yt−3, but not between yt and yt−s, for s > 3. Hence the pacf will usually have non-zero partial autocorrelation coefficients for lags up to the order of the model, but will have zero partial autocorrelation coefficients thereafter. In the case of the AR(3), only the first three partial autocorrelation coefficients will be non-zero. What shape would the partial autocorrelation function take for a moving average process? One would need to think about the MA model as being transformed into an AR in order to consider whether yt and yt−k, k = 1, 2, …, are directly connected. In fact, so long as the MA(q) process is invertible, it can be expressed as an AR(∞). Thus a definition of invertibility is now required.

6.5.1 The Invertibility Condition An MA(q) model is typically required to have roots of the characteristic equation θ(z) = 0 greater than one in absolute value. The invertibility condition is mathematically the same as the stationarity condition, but is different in the sense that the former refers to MA rather than AR processes. This condition prevents the model from exploding under an AR(∞) representation, so that θ−1(L) converges to zero. Box 6.2 shows the invertibility condition for an MA(2) model. BOX 6.2 The invertibility condition for an MA(2) model In order to examine the shape of the pacf for moving average processes, consider the following MA(2) process for yt (6.119) Provided that this process is invertible, this MA(2) can be expressed as an AR(∞) (6.120) (6.121)

350

It is now evident when expressed in this way that for a moving average model, there are direct connections between the current value of y and all of its previous values. Thus, the partial autocorrelation function for an MA(q) model will decline geometrically, rather than dropping off to zero after q lags, as is the case for its autocorrelation function. It could thus be stated that the acf for an AR has the same basic shape as the pacf for an MA, and the acf for an MA has the same shape as the pacf for an AR.

6.6 ARMA Processes By combining the AR(p) and MA(q) models, an ARMA(p, q) model is obtained. Such a model states that the current value of some series y depends linearly on its own previous values plus a combination of current and previous values of a white noise error term. The model could be written (6.122) where

or (6.123) with

The characteristics of an ARMA process will be a combination of those from the autoregressive (AR) and moving average (MA) parts. Note that the pacf is particularly useful in this context. The acf alone can distinguish between a pure autoregressive and a pure moving average process. However, an ARMA process will have a geometrically declining acf, as will a pure AR process. So, the pacf is useful for distinguishing between an AR(p) process and an ARMA(p, q) process – the former will have a 351

geometrically declining autocorrelation function, but a partial autocorrelation function which cuts off to zero after p lags, while the latter will have both autocorrelation and partial autocorrelation functions which decline geometrically. We can now summarise the defining characteristics of AR, MA and ARMA processes. An autoregressive process has: a geometrically decaying acf a number of non-zero points of pacf = AR order. A moving average process has: number of non-zero points of acf = MA order a geometrically decaying pacf. A combination autoregressive moving average process has: a geometrically decaying acf ageometricallydecayingpacf. In fact, the mean of an ARMA series is given by (6.124) The autocorrelation function will display combinations of behaviour derived from the AR and MA parts, but for lags beyond q, the acf will simply be identical to the individual AR(p) model, so that the AR part will dominate in the long term. Deriving the acf and pacf for an ARMA process requires no new algebra, but is tedious and hence is left as an exercise for interested readers.

6.6.1 Sample acf and pacf Plots for Standard Processes Figures 6.2–6.8 give some examples of typical processes from the ARMA family with their characteristic autocorrelation and partial autocorrelation functions. The acf and pacf are not produced analytically from the relevant formulae for a model of that type, but rather are estimated using 100,000 simulated observations with disturbances drawn from a normal distribution. Each figure also has 5% (two-sided) rejection bands 352

represented by dotted lines. These are based on (±1.96/√100000) = ±0.0062, calculated in the same way as given above. Notice how, in each case, the acf and pacf are identical for the first lag.

Figure 6.2 Sample autocorrelation and partial autocorrelation functions

for an MA(1) model: yt = −0.5ut−1 + ut In Figure 6.2, the MA(1) has an acf that is significant for only lag 1, while the pacf declines geometrically, and is significant until lag 7. The acf at lag 1 and all of the pacfs are negative as a result of the negative coefficient in the MA generating process. Again, the structures of the acf and pacf in Figure 6.3 are as anticipated. The first two autocorrelation coefficients only are significant, while the partial autocorrelation coefficients are geometrically declining. Note also that, since the second coefficient on the lagged error term in the MA is negative, the acf and pacf alternate between positive and negative. In the case of the pacf, we term this alternating and declining function a ‘damped sine wave’ or ‘damped sinusoid’.

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Figure 6.3 Sample autocorrelation and partial autocorrelation functions

for an MA(2) model: yt = 0.5ut−1 − 0.25ut−2 + ut For the autoregressive model of order 1 with a fairly high coefficient – i.e., relatively close to 1 – the autocorrelation function would be expected to die away relatively slowly, and this is exactly what is observed here in Figure 6.4. Again, as expected for an AR(1), only the first pacf coefficient is significant, while all others are virtually zero and are not significant.

Figure 6.4 Sample autocorrelation and partial autocorrelation functions

for a slowly decaying AR(1) model: yt = 0.9yt−1 + ut

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Figure 6.5 plots an AR(1), which was generated using identical error terms, but a much smaller autoregressive coefficient. In this case, the autocorrelation function dies away much more quickly than in the previous example, and in fact becomes insignificant after around five lags.

Figure 6.5 Sample autocorrelation and partial autocorrelation functions

for a more rapidly decaying AR(1) model: yt = 0.5yt−1 + ut Figure 6.6 shows the acf and pacf for an identical AR(1) process to that used for Figure 6.5, except that the autoregressive coefficient is now negative. This results in a damped sinusoidal pattern for the acf, which again becomes insignificant after around lag 5. Recalling that the autocorrelation coefficient for this AR(1) at lag s is equal to (−0.5)s, this will be positive for even s, and negative for odd s. Only the first pacf coefficient is significant (and negative).

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Figure 6.6 Sample autocorrelation and partial autocorrelation functions

for a more rapidly decaying AR(1) model with negative coefficient: yt = −0.5yt−1 + ut Figure 6.7 plots the acf and pacf for a non-stationary series (see Chapter 8 for an extensive discussion) that has a unit coefficient on the lagged dependent variable. The result is that shocks to y never die away, and persist indefinitely in the system. Consequently, the acf function remains relatively flat at unity, even up to lag 10. In fact, even by lag 10, the autocorrelation coefficient has fallen only to 0.9989. Note also that on some occasions, the acf does die away, rather than looking like Figure 6.7, even for such a non-stationary process, owing to its inherent instability combined with finite computer precision. The pacf, however, is significant only for lag 1, correctly suggesting that an autoregressive model with no moving average term is most appropriate.

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Figure 6.7 Sample autocorrelation and partial autocorrelation functions

for a non-stationary model (i.e., a unit coefficient): yt = yt−1 + ut Finally, Figure 6.8 plots the acf and pacf for a mixed ARMA process. As one would expect of such a process, both the acf and the pacf decline geometrically – the acf as a result of the AR part and the pacf as a result of the MA part. The coefficients on the AR and MA are, however, sufficiently small that both acf and pacf coefficients have become insignificant by lag 6.

Figure 6.8 Sample autocorrelation and partial autocorrelation functions

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for an ARMA(1, 1) model: yt = 0.5yt−1 + 0.5ut−1 + ut

6.7 Building ARMA Models: The Box–Jenkins Approach Although the existence of ARMA models predates them, Box and Jenkins (1976) were the first to approach the task of estimating an ARMA model in a systematic manner. Their approach was a practical and pragmatic one, involving three steps: (1) Identification (2) Estimation (3) Diagnostic checking. These steps are now explained in greater detail. Step 1 This involves determining the order of the model required to capture the dynamic features of the data. Graphical procedures are used (plotting the data over time and plotting the acf and pacf) to determine the most appropriate specification. Step 2 This involves estimation of the parameters of the model specified in step 1. This can be done using least squares or another technique, known as maximum likelihood, depending on the model. Step 3 This involves model checking – i.e., determining whether the model specified and estimated is adequate. Box and Jenkins suggest two methods: overfitting and residual diagnostics. Overfitting involves deliberately fitting a larger model than that required to capture the dynamics of the data as identified in stage 1. If the model specified at step 1 is adequate, any extra terms added to the ARMA model would be insignificant. Residual diagnostics imply checking the residuals for evidence of linear dependence which, if present, would suggest that the 358

model originally specified was inadequate to capture the features of the data. The acf, pacf or Ljung–Box tests could be used. It is worth noting that ‘diagnostic testing’ in the Box–Jenkins world essentially involves only autocorrelation tests rather than the whole barrage of tests outlined in Chapter 5. Also, such approaches to determining the adequacy of the model could only reveal a model that is underparameterised (‘too small’) and would not reveal a model that is overparameterised (‘too big’). Examining whether the residuals are free from autocorrelation is much more commonly used than overfitting, and this may partly have arisen since for ARMA models, it can give rise to common factors in the overfitted model that make estimation of this model difficult and the statistical tests ill behaved. For example, if the true model is an ARMA(1,1) and we deliberately then fit an ARMA(2,2) there will be a common factor so that not all of the parameters in the latter model can be identified. This problem does not arise with pure AR or MA models, only with mixed processes. It is usually the objective to form a parsimonious model, which is one that describes all of the features of data of interest using as few parameters (i.e., as simple a model) as possible. A parsimonious model is desirable because: The residual sum of squares is proportional to the number of degrees of freedom. A model which contains irrelevant lags of the variable or of the error term (and therefore unnecessary parameters) will usually lead to increased coefficient standard errors, implying that it will be more difficult to find significant relationships in the data. Whether an increase in the number of variables (i.e., a reduction in the number of degrees of freedom) will actually cause the estimated parameter standard errors to rise or fall will obviously depend on how much the RSS falls, and on the relative sizes of T and k. If T is very large relative to k, then the decrease in RSS is likely to outweigh the reduction in T − k so that the standard errors fall. Hence ‘large’ models with many parameters are more often chosen when the sample size is large. Models that are profligate might be inclined to fit to data specific features, which would not be replicated out-of-sample. This means that the models may appear to fit the data very well, with perhaps a high value of R2, but would give very inaccurate forecasts. Another interpretation of this concept, borrowed from physics, is that of the 359

distinction between ‘signal’ and ‘noise’. The idea is to fit a model which captures the signal (the important features of the data, or the underlying trends or patterns), but which does not try to fit a spurious model to the noise (the completely random aspect of the series).

6.7.1 Information Criteria for ARMA Model Selection The identification stage would now typically not be done using graphical plots of the acf and pacf. The reason is that when ‘messy’ real data are used, they unfortunately rarely exhibit the simple patterns of Figures 6.2–6.8. This makes the acf and pacf very hard to interpret, and thus it is difficult to specify a model for the data. Another technique, which removes some of the subjectivity involved in interpreting the acf and pacf, is to use what are known as information criteria. Information criteria embody two factors: a term which is a function of the residual sum of squares (RSS), and some penalty for the loss of degrees of freedom from adding extra parameters. So, adding a new variable or an additional lag to a model will have two competing effects on the information criteria: the residual sum of squares will fall but the value of the penalty term will increase. The object is to choose the number of parameters which minimises the value of the information criteria. So, adding an extra term will reduce the value of the criteria only if the fall in the residual sum of squares is sufficient to more than outweigh the increased value of the penalty term. There are several different criteria, which vary according to how stiff the penalty term is. The three most popular information criteria are Akaike’s (1974) information criterion (AIC), Schwarz’s (1978) Bayesian information criterion (SBIC) and the Hannan–Quinn criterion (HQIC). Algebraically, these are expressed, respectively, as (6.125) (6.126) (6.127) where is the residual variance (also equivalent to the residual sum of squares divided by the number of observations, T), k = p + q + 1 is the total number of parameters estimated and T is the sample size. The information criteria are actually minimised subject to i.e., an upper limit is 360

specified on the number of moving average and/or autoregressive terms that will be considered. It is worth noting that SBIC embodies a much stiffer penalty term than AIC, while HQIC is somewhere in between. The adjusted R2 measure can also be viewed as an information criterion, although it is a very soft one, which would typically select the largest models of all.

6.7.2 Which Criterion Should be Preferred if they Suggest Different Model Orders? SBIC is strongly consistent (but inefficient) and AIC is not consistent, but is generally more efficient. In other words, SBIC will asymptotically deliver the correct model order, while AIC will deliver on average too large a model, even with an infinite amount of data. On the other hand, the average variation in selected model orders from different samples within a given population will be greater in the context of SBIC than AIC. Overall, then, no criterion is definitely superior to others.

6.7.3 ARIMA Modelling ARIMA modelling, as distinct from ARMA modelling, has the additional letter ‘I’ in the acronym, standing for ‘integrated’. An integrated autoregressive process is one whose characteristic equation has a root on the unit circle. Typically researchers difference the variable as necessary and then build an ARMA model on those differenced variables. An ARMA(p, q) model in the variable differenced d times is equivalent to an ARIMA(p, d, q) model on the original data – see Chapter 8 for further details. For the remainder of this chapter, it is assumed that the data used in model construction are stationary, or have been suitably transformed to make them stationary. Thus only ARMA models will be considered further.

6.8 Examples of Time-Series Modelling in Finance 6.8.1 Covered and Uncovered Interest Parity The determination of the price of one currency in terms of another (i.e., the exchange rate) has received a great deal of empirical examination in the international finance literature. Of these, three hypotheses in particular are studied – covered interest parity (CIP), uncovered interest parity (UIP) and 361

purchasing power parity (PPP). The first two of these will be considered as illustrative examples in this chapter, while PPP will be discussed in Chapter 8. All three relations are relevant for students of finance, for violation of one or more of the parities may offer the potential for arbitrage, or at least will offer further insights into how financial markets operate. All are discussed briefly here; for a more comprehensive treatment, see Cuthbertson and Nitzsche (2004) or the many references therein.

6.8.2 Covered Interest Parity Stated in its simplest terms, CIP implies that, if financial markets are efficient, it should not be possible to make a riskless profit by borrowing at a risk-free rate of interest in a domestic currency, switching the funds borrowed into another (foreign) currency, investing them there at a riskfree rate and locking in a forward sale to guarantee the rate of exchange back to the domestic currency. Thus, if CIP holds, it is possible to write (6.128) where ft and st are the log of the forward and spot prices of the domestic in terms of the foreign currency at time t (i.e., the exchange rates), r is the domestic interest rate and r* is the foreign interest rate. This is an equilibrium condition which must hold otherwise there would exist riskless arbitrage opportunities, and the existence of such arbitrage would ensure that any deviation from the condition cannot hold indefinitely. It is worth noting that, underlying CIP are the assumptions that the risk-free rates are truly risk-free – that is, there is no possibility for default risk. It is also assumed that there are no transactions costs, such as broker’s fees, bid–ask spreads, stamp duty, etc., and that there are no capital controls, so that funds can be moved without restriction from one currency to another.

6.8.3 Uncovered Interest Parity UIP takes CIP and adds to it a further condition known as ‘forward rate unbiasedness’ (FRU). Forward rate unbiasedness states that the forward rate of foreign exchange should be an unbiased predictor of the future value of the spot rate. If this condition does not hold, again in theory riskless arbitrage opportunities could exist. UIP, in essence, states that the expected change in the exchange rate should be equal to the interest rate 362

differential between that available risk-free in each of the currencies. Algebraically, this may be stated as (6.129) where the notation is as above with s and se being the spot exchange rate, r and r* the interest rates and is the expectation, made at time t of the spot exchange rate that will prevail at time t + 1. The literature testing CIP and UIP is huge with literally hundreds of published papers. Tests of CIP unsurprisingly (for it is a pure arbitrage condition) tend not to reject the hypothesis that the condition holds. Taylor (1987, 1989) has conducted extensive examinations of CIP, and concluded that there were historical periods when arbitrage was profitable, particularly during periods where the exchange rates were under management. Relatively simple tests of UIP and FRU take equations of the form (6.129) and add intuitively relevant additional terms. If UIP holds, these additional terms should be insignificant. Ito (1988) tests UIP for the yen/dollar exchange rate with the three-month forward rate for January 1973 until February 1985. The sample period is split into three as a consequence of perceived structural breaks in the series. Strict controls on capital movements were in force in Japan until 1977, when some were relaxed and finally removed in 1980. A Chow test confirms Ito’s intuition and suggests that the three sample periods should be analysed separately. Two separate regressions are estimated for each of the three sample subperiods (6.130) where st+3 is the spot interest rate prevailing at time t + 3, ft,3 is the forward rate for three periods ahead available at time t, and so on, and ut is an error term. A natural joint hypothesis to test is H0: a = 0 and b1 =0 and b2 = 0. This hypothesis represents the restriction that the deviation of the forward rate from the realised rate should have a mean value insignificantly different from zero (a = 0) and it should be independent of any information available at time t (b1 = 0 and b2 = 0). All three of these conditions must be fulfilled for UIP to hold. The second equation that Ito tests is (6.131) 363

where vt is an error term and the hypothesis of interest in this case is H0: a = 0 and b = 0. Equation (6.130) tests whether past forecast errors have information useful for predicting the difference between the actual exchange rate at time t + 3, and the value of it that was predicted by the forward rate. Equation (6.131) tests whether the forward premium has any predictive power for the difference between the actual exchange rate at time t + 3, and the value of it that was predicted by the forward rate. The results for the three sample periods are presented in Ito’s Table 3, and are adapted and reported here in Table 6.1. Table 6.1 Uncovered interest parity test results

Source: Ito (1988). Reprinted with permission from MIT Press Journals.

The main conclusion is that UIP clearly failed to hold throughout the period of strictest controls, but there is less and less evidence against UIP as controls were relaxed.

6.9 Exponential Smoothing Exponential smoothing is another modelling technique (not based on the ARIMA approach) that uses only a linear combination of the previous 364

values of a series for modelling it and for generating forecasts of its future values. Given that only previous values of the series of interest are used, the only question remaining is how much weight should be attached to each of the previous observations. Recent observations would be expected to have the most power in helping to forecast future values of a series. If this is accepted, a model that places more weight on recent observations than those further in the past would be desirable. On the other hand, observations a long way in the past may still contain some information useful for forecasting future values of a series, which would not be the case under a centred moving average. An exponential smoothing model will achieve this, by imposing a geometrically declining weighting scheme on the lagged values of a series. The equation for the model is (6.132) where α is the smoothing constant, with 0 < α < 1, yt is the current realised value, St is the current smoothed value. Since α + (1 − α) = 1, St is modelled as a weighted average of the current observation yt and the previous smoothed value. The model above can be rewritten to express the exponential weighting scheme more clearly. By lagging equation (6.132) by one period, the following expression is obtained (6.133) and lagging again (6.134) Substituting into (6.132) for St−1 from equation (6.133) (6.135) (6.136) Substituting into (6.136) for St−2 from equation (6.134) (6.137) (6.138)

365

T successive substitutions of this kind would lead to (6.139) Since α > 0, the effect of each observation declines geometrically as the variable moves another observation forward in time. In the limit as T → ∞, the second term tends to zero, so that the current smoothed value is a geometrically weighted infinite sum of the previous realisations. The forecasts from an exponential smoothing model are simply set to the current smoothed value, for any number of steps ahead, s (6.140) The exponential smoothing model can be seen as a special case of a Box– Jenkins model, an ARIMA(0,1,1), with MA coefficient (1 − α) – see Granger and Newbold (1986, p. 174). The technique above is known as single or simple exponential smoothing, and it can be modified to allow for trends (Holt’s method) or to allow for seasonality (Winter’s method) in the underlying variable. These augmented models are not pursued further in this text since there is a much better way to model the trends (using a unit root process – see Chapter 8) and the seasonalities (see Chapter 10) of the form that are typically present in financial data. Exponential smoothing has several advantages over the slightly more complex ARMA class of models discussed above. First, exponential smoothing is obviously very simple to use. There is no decision to be made on how many parameters to estimate (assuming only single exponential smoothing is considered). Thus it is easy to update the model if a new realisation becomes available. Among the disadvantages of exponential smoothing is the fact that it is overly simplistic and inflexible. Exponential smoothing models can be viewed as but one model from the ARIMA family, which may not necessarily be optimal for capturing any linear dependence in the data. Also, the forecasts from an exponential smoothing model do not converge on the long-term mean of the variable as the horizon increases. The upshot is that long-term forecasts are overly affected by recent events in the history of the series under investigation and will therefore be sub-optimal.

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6.10 Forecasting in Econometrics Although the words ‘forecasting’ and ‘prediction’ are sometimes given different meanings in some studies, in this text the words will be used synonymously. In this context, prediction or forecasting simply means an attempt to determine the values that a series is likely to take. Of course, forecasts might also usefully be made in a cross-sectional environment. Although the discussion below refers to time-series data, some of the arguments will carry over to the cross-sectional context. Determining the forecasting accuracy of a model is an important test of its adequacy. Some econometricians would go as far as to suggest that the statistical adequacy of a model in terms of whether it violates the CLRM assumptions or whether it contains insignificant parameters, is largely irrelevant if the model produces accurate forecasts. The following subsections of the book discuss why forecasts are made, how they are made from several important classes of models, how to evaluate the forecasts, and so on.

6.10.1 Why Forecast? Forecasts are made essentially because they are useful! Financial decisions often involve a long-term commitment of resources, the returns to which will depend upon what happens in the future. In this context, the decisions made today will reflect forecasts of the future state of the world, and the more accurate those forecasts are, the more utility (or money!) is likely to be gained from acting on them. Some examples in finance of where forecasts from econometric models might be useful include: Forecasting tomorrow’s return on a particular share Forecasting the price of a house given its characteristics Forecasting the riskiness of a portfolio over the next year Forecasting the volatility of bond returns Forecasting the correlation between US and UK stock market movements tomorrow Forecasting the likely number of defaults on a portfolio of home loans. Again, it is evident that forecasting can apply either in a cross-sectional or a time-series context. It is useful to distinguish between two approaches to 367

forecasting: Econometric (structural) forecasting – relates a dependent variable to one or more independent variables. Such models often work well in the long run, since a long-run relationship between variables often arises from no-arbitrage or market efficiency conditions. Examples of such forecasts would include return predictions derived from arbitrage pricing models, or long-term exchange rate prediction based on purchasing power parity or uncovered interest parity theory. Time series forecasting – involves trying to forecast the future values of a series given its previous values and/or previous values of an error term. The distinction between the two types is somewhat blurred – for example, it is not clear where vector autoregressive (VAR) models (see Chapter 7 for an extensive overview) fit into this classification. It is also worth distinguishing between point and interval forecasts. Point forecasts predict a single value for the variable of interest, while interval forecasts provide a range of values in which the future value of the variable is expected to lie with a given level of confidence.

6.10.2 The Difference Between In-Sample and Out-of-Sample Forecasts In-sample forecasts are those generated for the same set of data that was used to estimate the model’s parameters. One would expect the ‘forecasts’ of a model to be relatively good in-sample, for this reason. Therefore, a sensible approach to model evaluation through an examination of forecast accuracy is not to use all of the observations in estimating the model parameters, but rather to hold some observations back. The latter sample, sometimes known as a holdout sample, would be used to construct out-ofsample forecasts. To give an illustration of this distinction, suppose that some monthly FTSE returns for 120 months (January 1990–December 1999) are available. It would be possible to use all of them to build the model (and generate only in-sample forecasts), or some observations could be kept back, as shown in Figure 6.9.

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Figure 6.9 Use of in-sample and out-of-sample periods for analysis

What would be done in this case would be to use data from 1990M1 until 1998M12 to estimate the model parameters, and then the observations for 1999 would be forecast from the estimated parameters. Of course, where each of the in-sample and out-of-sample periods should start and finish is somewhat arbitrary and at the discretion of the researcher. One could then compare how close the forecasts for the 1999 months were relative to their actual values that are in the holdout sample. This procedure would represent a better test of the model than an examination of the in-sample fit of the model since the information from 1999M1 onwards has not been used when estimating the model parameters.

6.10.3 Some More Terminology: One-Step-Ahead versus Multi-Step-Ahead Forecasts and Rolling versus Recursive Samples A one-step-ahead forecast is a forecast generated for the next observation only, whereas multi-step-ahead forecasts are those generated for 1, 2, 3, …, s steps ahead, so that the forecasting horizon is for the next s periods. Whether one-step- or multi-step-ahead forecasts are of interest will be determined by the forecasting horizon of interest to the researcher. Suppose that the monthly FTSE data are used as described in the example above. If the in-sample estimation period stops in December 1998, then up to twelve-step-ahead forecasts could be produced, giving twelve predictions that can be compared with the actual values of the series. Comparing the actual and forecast values in this way is not ideal, for the forecasting horizon is varying from one to twelve steps ahead. It might be the case, for example, that the model produces very good forecasts for short horizons (say, one or two steps), but that it produces inaccurate forecasts further ahead. It would not be possible to evaluate whether this was in fact the case or not since only a single one-step-ahead forecast, a single two-step-ahead forecast, and so on, are available. An evaluation of the forecasts would require a considerably larger holdout sample. 369

A useful way around this problem is to use a recursive or rolling window, which generates a series of forecasts for a given number of steps ahead. A recursive forecasting model would be one where the initial estimation date is fixed, but additional observations are added one at a time to the estimation period. A rolling window, on the other hand, is one where the length of the in-sample period used to estimate the model is fixed, so that the start date and end date successively increase by one observation. Suppose now that only one-, two-, and three-step-ahead forecasts are of interest. They could be produced using the following recursive and rolling window approaches: Objective: to produce

Data used to estimate model parameters

1-, 2-, 3-step-ahead forecasts for:

Rolling window

Recursive window

1999M1, M2, M3

1990M1–1998M12

1990M1–1998M12

1999M2, M3, M4

1990M2–1999M1

1990M1–1999M1

1999M3, M4, M5

1990M3–1999M2

1990M1–1999M2

1999M4, M5, M6

1990M4–1999M3

1990M1–1999M3

1999M5, M6, M7

1990M5–1999M4

1990M1–1999M4

1999M6, M7, M8

1990M6–1999M5

1990M1–1999M5

1999M7, M8, M9

1990M7–1999M6

1990M1–1999M6

1999M8, M9, M10

1990M8–1999M7

1990M1–1999M7

1999M9, M10, M11

1990M9–1999M8

1990M1–1999M8

1999M10, M11, M12

1990M10–1999M9

1990M1–1999M9

The sample length for the rolling windows above is always set at 108 observations, while the number of observations used to estimate the parameters in the recursive case increases as we move down the table and through the sample.

6.10.4 Forecasting with Time-Series versus Structural Models To understand how to construct forecasts, the idea of conditional expectations is required. A conditional expectation would be expressed as 370

This expression states that the expected value of y is taken for time t + 1, conditional upon, or given, (|) all information available up to and including time t (Ωt). Contrast this with the unconditional expectation of y, which is the expected value of y without any reference to time, i.e., the unconditional mean of y. The conditional expectations operator is used to generate forecasts of the series. How this conditional expectation is evaluated will of course depend on the model under consideration. Several families of models for forecasting will be developed in this and subsequent chapters. A first point to note is that by definition the optimal forecast for a zero mean white noise process is zero (6.141) The two simplest forecasting ‘methods’ that can be employed in almost every situation are shown in Box 6.3. BOX 6.3 Naive forecasting methods (1) Assume no change so that the forecast, f, of the value of y, s steps into the future is the current value of y (6.142) Such a forecast would be optimal if yt followed a random walk process. (2) In the absence of a full model, forecasts can be generated using the long-term average of the series. Forecasts using the unconditional mean would be more useful than ‘no change’ forecasts for any series that is ‘mean-reverting’ (i.e., stationary). Time series models are generally better suited to the production of time series forecasts than structural models. For an illustration of this, consider the following linear regression model (6.143)

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To forecast y, the conditional expectation of its future value is required. We take conditional expectations of both sides of equation (6.143), and note that strictly conditional expectations should be added to all variables on the RHS of equations (6.144) and (6.145) (6.144) The parameters can be taken through the expectations operator, since this is a population regression function and therefore they are assumed known. The following expression would be obtained (6.145) But there is a problem: what are E(x2t), etc.? Remembering that information is available only until time t − 1, the values of these variables are unknown. It may be possible to forecast them, but this would require another set of forecasting models for every explanatory variable. To the extent that forecasting the explanatory variables may be as difficult, or even more difficult, than forecasting the explained variable, this equation has achieved nothing! In the absence of a set of forecasts for the explanatory variables, one might think of using etc., i.e., the mean values of the explanatory variables, giving (6.146) Thus, if the mean values of the explanatory variables are used as inputs to the model, all that will be obtained as a forecast is the average value of y. Forecasting using pure time series models is relatively common, since it avoids this problem.

6.10.5 Forecasting with ARMA Models Forecasting using ARMA models is a fairly simple exercise in calculating conditional expectations. Although any consistent and logical notation could be used, the following conventions will be adopted in this book. Let ft,s denote a forecast made using an ARMA(p,q) model at time t for s steps into the future for some series y. The forecasts are generated by what is known as a forecast function, typically of the form

372

(6.147)

where ft,s = yt+s, s ≤ 0; ut+s = 0, s > 0 = ut+s, s ≤ 0 and ai and bi are the autoregressive and moving average coefficients, respectively. A demonstration of how one generates forecasts for separate AR and MA processes, leading to the general equation (6.147) above, will now be given.

6.10.6 Forecasting the Future Value of an MA(q) Process A moving average process has a memory only of length q, and this limits the sensible forecasting horizon. For example, suppose that an MA(3) model has been estimated (6.148) Since parameter constancy over time is assumed, if this relationship holds for the series y at time t, it is also assumed to hold for y at time t + 1, t + 2, …, so 1 can be added to each of the time subscripts in equation (6.148), and 2 added to each of the time subscripts, and then 3, and so on, to arrive at the following (6.149) (6.150) (6.151) Suppose that all information up to and including that at time t is available and that forecasts for 1, 2, …, s steps ahead – i.e., forecasts for y at times t + 1, t + 2, …, t + s are wanted. yt, yt−1, …, and ut, ut−1, are known, so producing the forecasts is just a matter of taking the conditional expectation of equation (6.149) (6.152) where E(yt+1|t) is a short-hand notation for E(yt+1|Ωt) (6.153) 373

Thus the forecast for y, one step ahead, made at time t, is given by this linear combination of the disturbance terms. Note that it would not be appropriate to set the values of these disturbance terms to their unconditional mean of zero. This arises because it is the conditional expectation of their values that is of interest. Given that all information is known up to and including that at time t is available, the values of the error terms up to time t are known. But ut+1 is not known at time t and therefore E(ut+1|t) = 0, and so on. The forecast for two steps ahead is formed by taking the conditional expectation of equation (6.150) (6.154) (6.155) In the case above, ut+2 is not known since information is available only to time t, so E(ut+2) is set to zero. Continuing and applying the same rules to generate 3-, 4-, …, s-step-ahead forecasts (6.156) (6.157) (6.158) (6.159) As the MA(3) process has a memory of only three periods, all forecasts four or more steps ahead collapse to the intercept. Obviously, if there had been no constant term in the model, the forecasts four or more steps ahead for an MA(3) would be zero.

6.10.7 Forecasting the Future Value of an AR(p) Process Unlike a moving average process, an autoregressive process has infinite memory. To illustrate, suppose that an AR(2) model has been estimated (6.160) Again, by appealing to the assumption of parameter stability, this equation will hold for times t + 1, t + 2, and so on (6.161) 374

(6.162) (6.163) Producing the one-step-ahead forecast is easy, since all of the information required is known at time t. Applying the expectations operator to equation (6.161), and setting E(ut+1) to zero would lead to (6.164) (6.165) (6.166) Applying the same procedure in order to generate a two-step-ahead forecast (6.167) (6.168) The case above is now slightly more tricky, since E(yt+1) is not known, although this in fact is the one-step-ahead forecast, so that equation (6.168) becomes (6.169) Similarly, for three, four, …and s steps ahead, the forecasts will be, respectively, given by (6.170) (6.171) (6.172) (6.173) etc. so (6.174) Thus the s-step-ahead forecast for an AR(2) process is given by the intercept + the coefficient on the one-period lag multiplied by the time s − 1 forecast + the coefficient on the two-period lag multiplied by the s − 2 forecast. 375

ARMA(p,q) forecasts can easily be generated in the same way by applying the rules for their component parts, and using the general formula given by equation (6.147).

6.10.8 Determining Whether a Forecast is Accurate or Not For example, suppose that tomorrow’s return on the FTSE is predicted to be 0.2, and that the outcome is actually −0.4. Is this an accurate forecast? Clearly, one cannot determine whether a forecasting model is good or not based upon only one forecast and one realisation. Thus in practice, forecasts would usually be produced for the whole of the out-of-sample period, which would then be compared with the actual values, and the difference between them aggregated in some way. The forecast error for observation i is defined as the difference between the actual value for observation i and the forecast made for it. The forecast error, defined in this way, will be positive (negative) if the forecast was too low (high). Therefore, it is not possible simply to sum the forecast errors, since the positive and negative errors will cancel one another out. Thus, before the forecast errors are aggregated, they are usually squared or the absolute value taken, which renders them all positive. To see how the aggregation works, consider the example in Table 6.2, where forecasts are made for a series up to five steps ahead, and are then compared with the actual realisations (with all calculations rounded to three decimal places). Table 6.2 Forecast error aggregation Steps ahead

Forecast

Actual

Squared error

Absolute error

1

0.20

−0.40

(0.20 − −0.40)2 = 0.360

|0.20 − −0.40| = 0.600

2

0.15

0.20

(0.15−0.20)2 = 0.002

|0.15−0.20| = 0.050

3

0.10

0.10

(0.10−0.10)2 = 0.000

|0.10−0.10| = 0.000

4

0.06

−0.10

(0.06 − −0.10)2 = 0.026

|0.06 − −0.10| = 0.160

5

0.04

−0.05

(0.04 − −0.05)2 = 0.008

|0.06 − −0.10| = 0.160

376

The mean squared error (MSE) and mean absolute error (MAE) are now calculated by taking the average of the fourth and fifth columns, respectively (6.175) (6.176) Taken individually, little can be gleaned from considering the size of the MSE or MAE, for the statistic is unbounded from above (like the residual sum of squares or RSS). Instead, the MSE or MAE from one model would be compared with those of other models for the same data and forecast period, and the model(s) with the lowest value of the error measure would be argued to be the most accurate. MSE provides a quadratic loss function, and so may be particularly useful in situations where large forecast errors are disproportionately more serious than smaller errors. This may, however, also be viewed as a disadvantage if large errors are not disproportionately more serious, although the same critique could also, of course, be applied to the whole least squares methodology. Indeed Dielman (1986) goes as far as to say that when there are outliers present, least absolute values should be used to determine model parameters rather than least squares. Makridakis (1993, p. 528) argues that mean absolute percentage error (MAPE) is ‘a relative measure that incorporates the best characteristics among the various accuracy criteria’. Once again, denoting s-step-ahead forecasts of a variable made at time t as ft,s and the actual value of the variable at time t as yt, then the MSE can be defined as (6.177) where T is the total sample size (in-sample + out-of-sample), and T1 is the first outof-sample forecast observation. Thus in-sample model estimation initially runs from observation 1 to (T1−1), and observations T1 to T are available for out-of-sample estimation, i.e., a total holdout sample of T − (T1 − 1). MAE measures the average absolute forecast error, and is given by

377

(6.178)

Adjusted MAPE (AMAPE) or symmetric MAPE corrects for the problem of asymmetry between the actual and forecast values (6.179) The symmetry in equation (6.179) arises since the forecast error is divided by twice the average of the actual and forecast values. So, for example, AMAPE will be the same whether the forecast is 0.5 and the actual value is 0.3, or the actual value is 0.5 and the forecast is 0.3. The same is not true of the standard MAPE formula, where the denominator is simply yt+s, so that whether yt or ft,s is larger will affect the result (6.180) MAPE also has the attractive additional property compared to MSE that it can be interpreted as a percentage error, and furthermore, its value is bounded from below by 0. Unfortunately, it is not possible to use the adjustment if the series and the forecasts can take on opposite signs (as they could in the context of returns forecasts, for example). This is due to the fact that the prediction and the actual value may, purely by coincidence, take on values that are almost equal and opposite, thus almost cancelling each other out in the denominator. This leads to extremely large and erratic values of AMAPE. In such an instance, it is not possible to use MAPE as a criterion either. Consider the following example: say we forecast a value of ft,s = 3, but the out-turn is that yt+s = 0.0001. The addition to total MSE from this one observation is given by (6.181) This value for the forecast is large, but perfectly feasible since in many cases it will be well within the range of the data. But the addition to total MAPE from just this single observation is given by

378

(6.182) MAPE has the advantage that for a random walk in the log levels (i.e., a zero forecast), the criterion will take the value one (or 100 if we multiply the formula by 100 to get a percentage, as was the case for the equation above). So if a forecasting model gives a MAPE smaller than one (or 100), it is superior to the random walk model. In fact the criterion is also not reliable if the series can take on absolute values less than one. This point may seem somewhat obvious, but it is clearly important for the choice of forecast evaluation criteria. Another criterion which is popular is Theil’s U-statistic (1966). The metric is defined as follows

(6.183)

where fbt,s is the forecast obtained from a benchmark model (typically a simple model such as a naive or random walk). A U-statistic of one implies that the model under consideration and the benchmark model are equally (in)accurate, while a value of less than one implies that the model is superior to the benchmark, and vice versa for U > 1. Although the measure is clearly useful, as Makridakis and Hibon (1995) argue, it is not without problems since if fbt,s is the same as yt+s, U will be infinite since the denominator will be zero. The value of Uwill also be influenced by outliers in a similar vein to MSE and has little intuitive meaning.2

6.10.9 Statistical versus Financial or Economic Loss Functions Many econometric forecasting studies evaluate the models’ success using statistical loss functions such as those described above. However, it is not necessarily the case that models classed as accurate because they have small mean squared forecast errors are useful in practical situations. To give one specific illustration, it has been shown (Gerlow, Irwin and Liu, 1993) that the accuracy of forecasts according to traditional statistical criteria may give little guide to the potential profitability of employing those forecasts in a market trading strategy. So models that perform poorly on statistical grounds may still yield a profit if used for trading, and vice 379

versa. On the other hand, models that can accurately forecast the sign of future returns, or can predict turning points in a series have been found to be more profitable (Leitch and Tanner, 1991). Two possible indicators of the ability of a model to predict direction changes irrespective of their magnitude are those suggested by Pesaran and Timmerman (1992) and by Refenes (1995). The relevant formulae to compute these measures are, respectively, (6.184)

and (6.185)

Thus, in each case, the criteria give the proportion of correctly predicted signs and directional changes for some given lead time s, respectively. Considering how strongly each of the three criteria outlined above (MSE, MAE and proportion of correct sign predictions) penalises large errors relative to small ones, the criteria can be ordered as follows

MSE penalises large errors disproportionately more heavily than small errors, MAE penalises large errors proportionately equally as heavily as small errors, while the sign prediction criterion does not penalise large errors any more than small errors.

6.10.10 Finance Theory and Time-Series Analysis An example of ARIMA model identification, estimation and forecasting in the context of commodity prices is given by Chu (1978). He finds ARIMA 380

models useful compared with structural models for short-term forecasting, but also finds that they are less accurate over longer horizons. It also observed that ARIMA models have limited capacity to forecast unusual movements in prices. Chu (1978) argues that, although ARIMA models may appear to be completely lacking in theoretical motivation, and interpretation, this may not necessarily be the case. He cites several papers and offers an additional example to suggest that ARIMA specifications quite often arise naturally as reduced form equations (see Chapter 7) corresponding to some underlying structural relationships. In such a case, not only would ARIMA models be convenient and easy to estimate, they could also be well grounded in financial or economic theory after all. KEY CONCEPTS The key terms to be able to define and explain from this chapter are ARIMA models invertible MA autocorrelation function Box–Jenkins methodology exponential smoothing rolling window multi-step forecast mean absolute percentage error Ljung–Box test Wold’s decomposition theorem partial autocorrelation function information criteria recursive window out-of-sample mean squared error

SELF-STUDY QUESTIONS 1. What are the differences between autoregressive and moving average models? 2. Why might ARMA models be considered particularly useful for 381

financial time series? Explain, without using any equations or mathematical notation, the difference between AR, MA and ARMA processes. 3. Consider the following three models that a researcher suggests might be a reasonable model of stock market prices

(a) (b)

What classes of models are these examples of? What would the autocorrelation function for each of these processes look like? (You do not need to calculate the acf, simply consider what shape it might have given the class of model from which it is drawn.) (c) Which model is more likely to represent stock market prices from a theoretical perspective, and why? If any of the three models truly represented the way stock market prices move, which could potentially be used to make money by forecasting future values of the series? (d) By making a series of successive substitutions or from your knowledge of the behaviour of these types of processes, consider the extent of persistence of shocks in the series in each case. 4. (a) Describe the steps that Box and Jenkins (1976) suggested should be involved in constructing an ARMA model. (b) What particular aspect of this methodology has been the subject of criticism and why? (c) Describe an alternative procedure that could be used for this aspect. 5. You obtain the following estimates for an AR(2) model of some returns data where ut is a white noise error process. By examining the characteristic equation, check the estimated model for stationarity. 6. A researcher is trying to determine the appropriate order of an ARMA model to describe some actual data, with 200 observations 382

available. She has the following figures for the log of the estimated residual variance (i.e., ) for various candidate models. She has assumed that an order greater than (3,3) should not be necessary to model the dynamics of the data. What is the ‘optimal’ model order? ARMA(p,q) model order (0,0) (1,0) (0,1) (1,1) (2,1) (1,2) (2,2) (3,2) (2,3) (3,3)

0.932 0.864 0.902 0.836 0.801 0.821 0.789 0.773 0.782 0.764

7. How could you determine whether the order you suggested for Question 6 was in fact appropriate? 8. ‘Given that the objective of any econometric modelling exercise is to find the model that most closely ‘fits’ the data, then adding more lags to an ARMA model will almost invariably lead to a better fit. Therefore a large model is best because it will fit the data more closely.’ Comment on the validity (or otherwise) of this statement. 9. (a) You obtain the following sample autocorrelations and partial autocorrelations for a sample of 100 observations from actual data: Lag 1 2 3 4 5 6 7 acf 0.420 0.104 0.032 −0.206 −0.138 0.042 −0.018 pacf 0.632 0.381 0.268 0.199 0.205 0.101 0.096 Can you identify the most appropriate time-series process for this data? (b)

Use the Ljung–Box Q* test to determine whether the first 383

three autocorrelation coefficients taken together are jointly significantly different from zero. 10. You have estimated the following ARMA(1,1) model for some time-series data Suppose that you have data for time to t− 1, i.e., you know that yt −1 = 3.4, and (a) (b)

(c)

(d)

11. (a)

Obtain forecasts for the series y for times t, t + 1, and t + 2 using the estimated ARMA model. If the actual values for the series turned out to be −0.032, 0.961, 0.203 for t, t + 1, t + 2, calculate the (out-of-sample) mean squared error. A colleague suggests that a simple exponential smoothing model might be more useful for forecasting the series. The estimated value of the smoothing constant is 0.15, with the most recently available smoothed value, St−1 being 0.0305. Obtain forecasts for the series y for times t, t +1, and t +2 using this model. Given your answers to parts (a) to (c) of the question, determine whether Box–Jenkins or exponential smoothing models give the most accurate forecasts in this application. Explain what stylised shapes would be expected for the autocorrelation and partial autocorrelation functions for the following stochastic processes: • • • •

(b)

white noise an AR(2) an MA(1) an ARMA (2,1).

Consider the following ARMA process.

Determine whether the MA part of the process is invertible. (c) Produce one-, two-, three- and four-step-ahead forecasts for the process given in part (b) of the question. (d) Outline two criteria that are available for evaluating the 384

forecasts produced in part (c) of the question, highlighting the differing characteristics of each. (e) What procedure might be used to estimate the parameters of an ARMA model? Explain, briefly, how such a procedure operates, and why OLS is not appropriate. 12. (a) Briefly explain any difference you perceive between the characteristics of macroeconomic and financial data. Which of these features suggest the use of different econometric tools for each class of data? (b) Consider the following autocorrelation and partial autocorrelation coefficients estimated using 500 observations for a weakly stationary series, yt : Lag acf pacf 1 0.307 0.307 2 −0.013 0.264 3 0.086 0.147 4 0.031 0.086 5 −0.197 0.049 Using a simple ‘rule of thumb’, determine which, if any, of the acf and pacf coefficients are significant at the 5% level. Use both the Box–Pierce and Ljung–Box statistics to test the joint null hypothesis that the first five autocorrelation coefficients are jointly zero. (c) What process would you tentatively suggest could represent the most appropriate model for the series in part (b)? Explain your answer. (d) Two researchers are asked to estimate an ARMA model for a daily USD/GBP exchange rate return series, denoted xt. Researcher A uses Schwarz’s criterion for determining the appropriate model order and arrives at an ARMA(0,1). Researcher B uses Akaike’s information criterion which deems an ARMA(2,0) to be optimal. The estimated models are

385

where ut is an error term. You are given the following data for time until day z (i.e., t = z)

Produce forecasts for the next four days (i.e., for times z + 1, z + 2, z + 3, z + 4) from both models. (e) Outline two methods proposed by Box and Jenkins (1976) for determining the adequacy of the models proposed in part (d). (f) Suppose that the actual values of the series x on days z +1, z +2, z + 3, z + 4 turned out to be 0.62, 0.19, −0.32, 0.72, respectively. Determine which researcher’s model produced the most accurate forecasts.

1

2

Note that the τs will not follow an exact geometric sequence, but rather the absolute value of the τs is bounded by a geometric series. This means that the autocorrelation function does not have to be monotonically decreasing and may change sign. Note that the Theil’s U-formula reported by EViews and some other software packages is slightly different.

386

7 Multivariate Models

LEARNING OUTCOMES In this chapter, you will learn how to Compare and contrast single equation and systems-based approaches to building models Discuss the cause, consequence and solution to simultaneous equations bias Derive the reduced form equations from a structural model Describe several methods for estimating simultaneous equations models Explain the relative advantages and disadvantages of VAR modelling Determine whether an equation from a system is identified Estimate optimal lag lengths, impulse responses and variance decompositions Conduct Granger causality tests

7.1 Motivations All of the structural models that have been considered thus far have been single equations models of the form (7.1) One of the assumptions of the classical linear regression model (CLRM) is that the explanatory variables are non-stochastic, or fixed in repeated 387

samples. There are various ways of stating this condition, some of which are slightly more or less strict, but all of which have the same broad implication. It could also be stated that all of the variables contained in the X matrix are assumed to be exogenous – that is, their values are determined outside that equation. This is a rather simplistic working definition of exogeneity, although several alternatives are possible; this issue will be revisited later in the chapter. Another way to state this is that the model is ‘conditioned on’ the variables in X. As stated in Chapter 3, the X matrix is assumed not to have a probability distribution. Note also that causality in this model runs from X to y, and not vice versa, i.e., that changes in the values of the explanatory variables cause changes in the values of y, but that changes in the value of y will not impact upon the explanatory variables. On the other hand, y is an endogenous variable – that is, its value is determined by equation (7.1). The purpose of the first part of this chapter is to investigate one of the important circumstances under which the assumption presented above will be violated. The impact on the OLS estimator of such a violation will then be considered. To illustrate a situation in which such a phenomenon may arise, consider the following two equations that describe a possible model for the total aggregate (country-wide) supply of new houses (or any other physical asset). (7.2) (7.3) (7.4) where Q dt = quantity of new houses demanded at time t Q st = quantity of new houses supplied (built) at time t Pt = (average) price of new houses prevailing at time t St = price of a substitute (e.g., older houses) Tt = some variable embodying the state of housebuilding technology, ut and vt are error terms.

388

Equation (7.2) is an equation for modelling the demand for new houses, and equation (7.3) models the supply of new houses. Equation (7.4) is an equilibrium condition for there to be no excess demand (people willing and able to buy new houses but cannot) and no excess supply (constructed houses that remain empty owing to lack of demand). Assuming that the market always clears, that is, that the market is always in equilibrium, and dropping the time subscripts for simplicity, equations (7.2)–(7.4) can be written (7.5) (7.6) Equations (7.5) and (7.6) together comprise a simultaneous structural form of the model, or a set of structural equations. These are the equations incorporating the variables that economic or financial theory suggests should be related to one another in a relationship of this form. The point is that price and quantity are determined simultaneously (price affects quantity and quantity affects price). Thus, in order to sell more houses, everything else equal, the builder will have to lower the price. Equally, in order to obtain a higher price for each house, the builder should construct and expect to sell fewer houses. P and Q are endogenous variables, while S and T are exogenous. A set of reduced form equations corresponding to equations (7.5) and (7.6) can be obtained by solving equations (7.5) and (7.6) for P and for Q (separately). There will be a reduced form equation for each endogenous variable in the system. Solving for Q (7.7) Solving for P (7.8) Rearranging equation (7.7) (7.9) 389

(7.10) (7.11) Multiplying equation (7.8) through by βμ and rearranging (7.12) (7.13) (7.14) (7.15) Equations (7.11) and (7.15) are the reduced form equations for P and Q. They are the equations that result from solving the simultaneous structural equations given by equations (7.5) and (7.6). Notice that these reduced form equations have only exogenous variables on the RHS.

7.2 Simultaneous Equations Bias It would not be possible to estimate equations (7.5) and (7.6) validly using OLS, as they are clearly related to one another since they both contain P and Q, and OLS would require them to be estimated separately. But what would have happened if a researcher had estimated them separately using OLS? Both equations depend on P. One of the CLRM assumptions was that X and u are independent (where X is a matrix containing all the variables on the RHS of the equation), and given also the assumption that E(u) = 0, then E(X′u) = 0, i.e., the errors are uncorrelated with the explanatory variables. But it is clear from equation (7.11) that P is related to the errors in equations (7.5) and (7.6) – i.e., it is stochastic. So this assumption has been violated. What would be the consequences for the OLS estimator, if the simultaneity were ignored? Recall that (7.16) and that 390

(7.17) Replacing y in (7.16) with the RHS of equation (7.17) (7.18) so that (7.19) (7.20) Taking expectations, (7.21) (7.22) If the Xs are non-stochastic (i.e., if the assumption had not been violated), E[(X′X)–1X′u] = (X′X)–1X′E[u] = 0, which would be the case in a single equation system, so that in (7.22). The implication is that the OLS estimator, would be unbiased. But, if the equation is part of a system, then E[(X′X)–1X′u] ≠ 0, in general, so that the last term in equation (7.22) will not drop out, and so it can be concluded that application of OLS to structural equations which are part of a simultaneous system will lead to biased coefficient estimates. This is known as simultaneity bias or simultaneous equations bias. Is the OLS estimator still consistent, even though it is biased? No, in fact, the estimator is inconsistent as well, so that the coefficient estimates would still be biased even if an infinite amount of data were available, although proving this would require a level of algebra beyond the scope of this book.

7.3 So how can Simultaneous Equations Models be Validly Estimated? Taking equations (7.11) and (7.15), i.e., the reduced form equations, they can be rewritten as

391

(7.23) (7.24) where the π coefficients in the reduced form are simply combinations of the original coefficients, so that

Equations (7.23) and (7.24) can be estimated using OLS since all the RHS variables are exogenous, so the usual requirements for consistency and unbiasedness of the OLS estimator will hold (provided that there are no other misspecifications). Estimates of the πij coefficients would thus be obtained. But, the values of the π coefficients are probably not of much interest; what was wanted were the original parameters in the structural equations – α, β, γ, λ, μ, κ. The latter are the parameters whose values determine how the variables are related to one another according to financial or economic theory.

7.4 Can the Original Coefficients be Retrieved from the π s? The short answer to this question is ‘sometimes’, depending upon whether the equations are identified. Identification is the issue of whether there is enough information in the reduced form equations to enable the structural form coefficients to be calculated. Consider the following demand and supply equations (7.25) (7.26) It is impossible to tell which equation is which, so that if one simply observed some quantities of a good sold and the price at which they were sold, it would not be possible to obtain the estimates of α, β, λ and μ. This arises since there is insufficient information from the equations to estimate 392

four parameters. Only two parameters could be estimated here, although each would be some combination of demand and supply parameters, and so neither would be of any use. In this case, it would be stated that both equations are unidentified (or not identified or underidentified). Notice that this problem would not have arisen with equations (7.5) and (7.6) since they have different exogenous variables.

7.4.1 What Determines Whether an Equation is Identified or Not? Any one of three possible situations could arise, as shown in Box 7.1. How can it be determined whether an equation is identified or not? Broadly, the answer to this question depends upon how many and which variables are present in each structural equation. There are two conditions that could be examined to determine whether a given equation from a system is identified – the order condition and the rank condition The order condition – is a necessary but not sufficient condition for an equation to be identified. That is, even if the order condition is satisfied, the equation might not be identified. The rank condition – is a necessary and sufficient condition for identification. The structural equations are specified in a matrix form and the rank of a coefficient matrix of all of the variables excluded from a particular equation is examined. An examination of the rank condition requires some technical algebra beyond the scope of this text. BOX 7.1 Determining whether an equation is identified (1) An equation is unidentified, such as equations (7.25) or (7.26). In the case of an unidentified equation, structural coefficients cannot be obtained from the reduced form estimates by any means. (2) An equation is exactly identified (just identified), such as equations (7.5) or (7.6). In the case of a just identified equation, unique structural form coefficient estimates can be obtained by substitution from the reduced form equations. (3) If an equation is overidentified, more than one set of structural coefficients can be obtained from the reduced form. An example of this will be presented later in this chapter. 393

Even though the order condition is not sufficient to ensure identification of an equation from a system, the rank condition will not be considered further here. For relatively simple systems of equations, the two rules would lead to the same conclusions. Also, in fact, most systems of equations in economics and finance are overidentified, so that underidentification is not a big issue in practice.

7.4.2 Statement of the Order Condition There are a number of different ways of stating the order condition; that employed here is an intuitive one (taken from Ramanathan, 1995, p. 666, and slightly modified): Let G denote the number of structural equations. An equation is just identified if the number of variables excluded from an equation is G–1, where ‘excluded’ means the number of all endogenous and exogenous variables that are not present in this particular equation. If more than G– 1 are absent, it is overidentified. If less than G–1 are absent, it is not identified. One obvious implication of this rule is that equations in a system can have differing degrees of identification, as illustrated by Example 7.1. EXAMPLE 7.1 In the following system of equations, the Ys are endogenous, while the Xs are exogenous (with time subscripts suppressed). Determine whether each equation is overidentified, underidentified, or just identified. (7.27) (7.28) (7.29) In this case, there are G = 3 equations and 3 endogenous variables. Thus, if the number of excluded variables is exactly 2, the equation is just identified. If the number of excluded variables is more than 2, the equation is overidentified. If the number of excluded variables is less 394

than 2, the equation is not identified. The variables that appear in one or more of the three equations are Y1, Y2, Y3, X1, X2. Applying the order condition to equations (7.27)–(7.29): Equation (7.27): contains all variables, with none excluded, so that it is not identified Equation (7.28): has variables Y1 and X2 excluded, and so is just identified Equation (7.29): has variables Y1 and X1 excluded, and so is also just identified

7.5 Simultaneous Equations in Finance There are of course numerous situations in finance where a simultaneous equations framework is more relevant than a single equation model. Two illustrations from the market microstructure literature are presented later in this chapter, while another, drawn from the banking literature, will be discussed now. There has recently been much debate internationally, but especially in the UK, concerning the effectiveness of competitive forces in the banking industry. Governments and regulators express concern at the increasing concentration in the industry, as evidenced by successive waves of merger activity, and at the enormous profits that many banks made in the late 1990s and early twenty-first century. They argue that such profits result from a lack of effective competition. However, many (most notably, of course, the banks themselves!) suggest that such profits are not the result of excessive concentration or anti-competitive practices, but rather partly arise owing to recent world prosperity at that phase of the business cycle (the ‘profits won’t last’ argument) and partly owing to massive costcutting by the banks, given recent technological improvements. These debates have fuelled a resurgent interest in models of banking profitability and banking competition. One such model is employed by Shaffer and DiSalvo (1994) in the context of two banks operating in south central Pennsylvania. The model is given by (7.30)

395

(7.31) where i = 1, 2 are the two banks, q is bank output, Pt is the price of the output at time t, Yt is a measure of aggregate income at time t, Zt is the price of a substitute for bank activity at time t, the variable t represents a time trend, TRit is the total revenue of bank i at time t, wikt are the prices of input k (k = 1, 2, 3 for labour, bank deposits and physical capital) for bank i at time t and the u are unobservable error terms. The coefficient estimates are not presented here, but suffice to say that a simultaneous framework, with the resulting model estimated separately using annual time-series data for each bank, is necessary. Output is a function of price on the RHS of equation (7.30), while in equation (7.31), total revenue, which is a function of output on the RHS, is obviously related to price. Therefore, OLS is again an inappropriate estimation technique. Both of the equations in this system are overidentified, since there are only two equations, and the income, the substitute for banking activity and the trend terms are missing from equation (7.31), whereas the three input prices are missing from equation (7.30).

7.6 A Definition of Exogeneity Leamer (1985) defines a variable x as exogenous if the conditional distribution of y given x does not change with modifications of the process generating x. Although several slightly different definitions exist, it is possible to classify two forms of exogeneity – predeterminedness and strict exogeneity A predetermined variable is one that is independent of the contemporaneous and future errors in that equation A strictly exogenous variable is one that is independent of all contemporaneous, future and past errors in that equation.

7.6.1 Tests for Exogeneity How can a researcher tell whether variables really need to be treated as endogenous or not? In other words, financial theory might suggest that there should be a twoway relationship between two or more variables, but how can it be tested whether a simultaneous equations model is necessary in practice? 396

EXAMPLE 7.2 Consider again equations (7.27)–(7.29). Equation (7.27) contains Y2 and Y3 – but are separate equations required for them, or could the variables Y2 and Y3 be treated as exogenous variables (in which case, they would be called X3 and X4!)? This can be formally investigated using a Hausman test, which is calculated as shown in Box 7.2. BOX 7.2 Conducting a Hausman test for exogeneity (1) Obtain the reduced form equations corresponding to equations (7.27)–(7.29). The reduced form equations are obtained as follows. Note that there is not a unique path to finding these solutions and several different routes would eventually arrive at the same reduced form equations. Substituting in equation (7.28) for Y3 from equation (7.29): (7.32) (7.33) (7.34) (7.35) Equation (7.35) is the reduced form equation for Y2, since there are no endogenous variables on the RHS. Substituting in equation (7.27) for Y3 from equation (7.29) (7.36)

(7.37)

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(7.38) Substituting in equation (7.38) for Y2 from equation (7.35)

(7.39)

(7.40)

(7.41)

Equation (7.41) is the reduced form equation for Y1. Finally, to obtain the reduced form equation for Y3, substitute in equation (7.29) for Y2 from equation (7.35) and simplifying

(7.42)

So, the reduced form equations corresponding to equations (7.27)–(7.29) are, respectively, given by equations (7.41), (7.35) and (7.42). These three equations can also be expressed using πij for the coefficients, as discussed above (7.43) 398

(7.44) (7.45) Estimate the reduced form equations (7.43)–(7.45) using OLS, and obtain the fitted values, where the superfluous superscript1 denotes the fitted values from the reduced form estimation. (2) Run the regression corresponding to equation (7.27) – i.e., the structural form equation, at this stage ignoring any possible simultaneity. (3) Run the regression equation (7.27) again, but now also including the fitted values from the reduced form equations, as additional regressors (7.46) (4) Use an F-test to test the joint restriction that λ2 = 0, and λ3 = 0. If the null hypothesis is rejected, Y2 and Y3 should be treated as endogenous. If λ2 and λ3 are significantly different from zero, there is extra important information for modelling Y1 from the reduced form equations. On the other hand, if the null is not rejected, Y2 and Y3 can be treated as exogenous for Y1, and there is no useful additional information available for Y1 from modelling Y2 and Y3 as endogenous variables. Steps (2)–(4) would then be repeated for equations (7.28) and (7.29).

7.7 Triangular Systems Consider the following system of equations, with time subscripts omitted for simplicity (7.47) (7.48)

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(7.49) Assume that the error terms from each of the three equations are not correlated with each other. Can the equations be estimated individually using OLS? At first blush, an appropriate answer to this question might appear to be, ‘No, because this is a simultaneous equations system’. But consider the following: Equation (7.47): contains no endogenous variables, so X1 and X2 are not correlated with u1. So OLS can be used on equation (7.47). Equation (7.48): contains endogenous Y1 together with exogenous X1 and X2. OLS can be used on equation (7.48) if all the RHS variables in equation (7.48) are uncorrelated with that equation’s error term. In fact, Y1 is not correlated with u2 because there is no Y2 term in equation (7.47). So OLS can be used on equation (7.48). Equation (7.49): contains both Y1 and Y2; these are required to be uncorrelated with u3. By similar arguments to the above, equations (7.47) and (7.48) do not contain Y3. So OLS can be used on equation (7.49). This is known as a recursive or triangular system, which is really a special case – a set of equations that looks like a simultaneous equations system, but isn’t. In fact, there is not a simultaneity problem here, since the dependence is not bi-directional, for each equation it all goes one way.

7.8 Estimation Procedures for Simultaneous Equations Systems Each equation that is part of a recursive system can be estimated separately using OLS. But in practice, not many systems of equations will be recursive, so a direct way to address the estimation of equations that are from a true simultaneous system must be sought. In fact, there are potentially many methods that can be used, three of which – indirect least squares, two-stage least squares and instrumental variables – will be detailed here. Each of these will be discussed below.

7.8.1 Indirect Least Squares (ILS) Although it is not possible to use OLS directly on the structural equations, 400

it is possible to validly apply OLS to the reduced form equations. If the system is just identified, ILS involves estimating the reduced form equations using OLS, and then using them to substitute back to obtain the structural parameters. ILS is intuitive to understand in principle; however, it is not widely applied because (1) Solving back to get the structural parameters can be tedious. For a large system, the equations may be set up in a matrix form, and to solve them may therefore require the inversion of a large matrix. (2) Most simultaneous equations systems are overidentified, and ILS can be used to obtain coefficients only for just identified equations. For overidentified systems, ILS would not yield unique structural form estimates. ILS estimators are consistent and asymptotically efficient, but in general they are biased, so that in finite samples ILS will deliver biased structural form estimates. In a nutshell, the bias arises from the fact that the structural form coefficients under ILS estimation are transformations of the reduced form coefficients. When expectations are taken to test for unbiasedness, it is in general not the case that the expected value of a (nonlinear) combination of reduced form coefficients will be equal to the combination of their expected values (see Gujarati, 2003, for a proof).

7.8.2 Estimation of Just Identified and Overidentified Systems using 2SLS This technique is applicable for the estimation of overidentified systems, where ILS cannot be used. In fact, it can also be employed for estimating the coefficients of just identified systems, in which case the method would yield asymptotically equivalent estimates to those obtained from ILS. Two-stage least squares (2SLS or TSLS) is done in two stages Stage 1 Obtain and estimate the reduced form equations using OLS. Save the fitted values for the dependent variables. Stage 2 Estimate the structural equations using OLS, but replace any RHS endogenous variables with their stage 1 fitted values. EXAMPLE 7.3 Suppose that equations (7.27)–(7.29) are required. 2SLS would involve the following two steps: 401

Stage 1 Estimate the reduced form equations (7.43)–(7.45) individually by OLS and obtain the fitted values, and denote them where the superfluous superscript1 indicates that these are the fitted values from the first stage. Stage 2 Replace the RHS endogenous variables with their stage 1 estimated values (7.50) (7.51) (7.52) where and are the fitted values from the reduced form estimation. Now and will not be correlated with u1, will not be correlated with u2, and will not be correlated with u3. The simultaneity problem has therefore been removed. It is worth noting that the 2SLS estimator is consistent, but not unbiased.

In a simultaneous equations framework, it is still of concern whether the usual assumptions of the CLRM are valid or not, although some of the test statistics require modifications to be applicable in the systems context. Most econometrics packages will automatically make any required changes. To illustrate one potential consequence of the violation of the CLRM assumptions, if the disturbances in the structural equations are autocorrelated, the 2SLS estimator is not even consistent. The standard error estimates also need to be modified compared with their OLS counterparts (again, econometrics software will usually do this automatically), but once this has been done, the usual t-tests can be used to test hypotheses about the structural form coefficients. This modification arises as a result of the use of the reduced form fitted values on the RHS rather than actual variables, which implies that a modification to the error variance is required.

7.8.3 Instrumental Variables Broadly, the method of instrumental variables (IV) is another technique for parameter estimation that can be validly used in the context of a 402

simultaneous equations system. Recall that the reason that OLS cannot be used directly on the structural equations is that the endogenous variables are correlated with the errors. One solution to this would be not to use Y2 or Y3, but rather to use some other variables instead. These other variables should be (highly) correlated with Y2 and Y3, but not correlated with the errors – such variables would be known as instruments. Suppose that suitable instruments for Y2 and Y3, were found and denoted z2 and z3, respectively. The instruments are not used in the structural equations directly, but rather, regressions of the following form are run (7.53) (7.54) Obtain the fitted values from equations (7.53) and (7.54), and and replace Y2 and Y3 with these in the structural equation. It is typical to use more than one instrument per endogenous variable. If the instruments are the variables in the reduced form equations, then IV is equivalent to 2SLS, so that the latter can be viewed as a special case of the former.

7.8.4 What Happens if IV or 2SLS are Used Unnecessarily? In other words, suppose that one attempted to estimate a simultaneous system when the variables specified as endogenous were in fact independent of one another. The consequences are similar to those of including irrelevant variables in a singleequation OLS model. That is, the coefficient estimates will still be consistent, but will be inefficient compared to those that just used OLS directly.

7.8.5 Other Estimation Techniques There are, of course, many other estimation techniques available for systems of equations, including three-stage least squares (3SLS), full information maximum likelihood (FIML) and limited information maximum likelihood (LIML). Three-stage least squares provides a third step in the estimation process that allows for non-zero covariances between the error terms in the structural equations. It is asymptotically more efficient than 2SLS since the latter ignores any information that may be available concerning the error covariances (and also any additional 403

information that may be contained in the endogenous variables of other equations). Full information maximum likelihood involves estimating all of the equations in the system simultaneously using maximum likelihood (see Chapter 8 for a discussion of the principles of maximum likelihood estimation). Thus under FIML, all of the parameters in all equations are treated jointly, and an appropriate likelihood function is formed and maximised. Finally, limited information maximum likelihood involves estimating each equation separately by maximum likelihood. LIML and 2SLS are asymptotically equivalent. For further technical details on each of these procedures, see Greene (2002, Chapter 15). The following section presents an application of the simultaneous equations approach in finance to the joint modelling of bid–ask spreads and trading activity in the S&P100 index options market. Two related applications of this technique that are also worth examining are by Wang, Yau and Baptiste (1997) and by Wang and Yau (2000). The former employs a bivariate system to model trading volume and bid–ask spreads and they show using a Hausman test that the two are indeed simultaneously related and so must both be treated as endogenous variables and are modelled using 2SLS. The latter paper employs a trivariate system to model trading volume, spreads and intra-day volatility.

7.9 An Application of a Simultaneous Equations Approach to Modelling Bid–Ask Spreads and Trading Activity 7.9.1 Introduction One of the most rapidly growing areas of empirical research in finance is the study of market microstructure. This research is involved with issues such as price formation in financial markets, how the structure of the market may affect the way it operates, determinants of the bid–ask spread, and so on. One application of simultaneous equations methods in the market microstructure literature is a study by George and Longstaff (1993). Among other issues, this paper considers the questions Is trading activity related to the size of the bid–ask spread? How do spreads vary across options, and how is this related to the volume of contracts traded? ‘Across options’ in this case means for different maturities and strike prices for an option on a given 404

underlying asset. This chapter will now examine the George and Longstaff models, results and conclusions.

7.9.2 The Data The data employed by George and Longstaff (1993) comprise options prices on the S&P100 index, observed on all trading days during 1989. The S&P100 index has been traded on the Chicago Board Options Exchange (CBOE) since 1983 on a continuous open-outcry auction basis. The option price as used in the paper is defined as the average of the bid and the ask. The average bid and ask prices are calculated for each option during the time 2.00p.m.–2.15p.m. (US Central Standard Time) to avoid time-of-day effects, such as differences in behaviour at the open and the close of the market. The following are then dropped from the sample for that day to avoid any effects resulting from stale prices Any options that do not have bid and ask quotes reported during the fifteen minutes Any options with fewer than ten trades during the day. This procedure results in a total of 2,456 observations. A ‘pooled’ regression is conducted since the data have both time-series and crosssectional dimensions. That is, the data are measured every trading day and across options with different strikes and maturities, and the data are stacked in a single column for analysis.

7.9.3 How Might the Option Price/Trading Volume and the Bid–Ask Spread be Related? George and Longstaff argue that the bid–ask spread will be determined by the interaction of market forces. Since there are many market makers trading the S&P100 contract on the CBOE, the bid–ask spread will be set to just cover marginal costs. There are three components of the costs associated with being a market maker. These are administrative costs, inventory holding costs and ‘risk costs’. George and Longstaff (1993) consider three possibilities for how the bid–ask spread might be determined Market makers equalise spreads across options This is likely to be 405

the case if order-processing (administrative) costs make up the majority of costs associated with being a market maker. This could be the case since the CBOE charges market makers the same fee for each option traded. In fact, for every contract (100 options) traded, a CBOE fee of 9 cents and an Options Clearing Corporation (OCC) fee of 10 cents is levied on the firm that clears the trade. The spread might be a constant proportion of the option value This would be the case if the majority of the market maker’s cost is in inventory holding costs, since the more expensive options will cost more to hold and hence the spread would be set wider. Market makers might equalise marginal costs across options irrespective of trading volume This would occur if the riskiness of an unwanted position were the most important cost facing market makers. Market makers typically do not hold a particular view on the direction of the market – they simply try to make money by buying and selling. Hence, they would like to be able to offload any unwanted (long or short) positions quickly. But trading is not continuous, and in fact the average time between trades in 1989 was approximately five minutes. The longer market-makers hold an option, the higher the risk they face since the higher the probability that there will be a large adverse price movement. Thus options with low trading volumes would command higher spreads since it is more likely that the market-maker would be holding these options for longer. In a non-quantitative exploratory analysis, George and Longstaff (1993) find that, comparing across contracts with different maturities, the bid–ask spread does indeed increase with maturity (as the option with longer maturity is worth more) and with ‘moneyness’ (that is, an option that is deeper in the money has a higher spread than one which is less in the money). This is seen to be true for both call and put options.

7.9.4 The Influence of Tick-Size Rules on Spreads The CBOE limits the tick size (the minimum granularity of price quotes), which will of course place a lower limit on the size of the spread. The tick sizes are $1/8 for options worth $3 or more $1/16 for options worth less than $3. 406

7.9.5 The Models and Results The intuition that the bid–ask spread and trading volume may be simultaneously related arises since a wider spread implies that trading is relatively more expensive so that marginal investors would withdraw from the market. On the other hand, market-makers face additional risk if the level of trading activity falls, and hence they may be expected to respond by increasing their fee (the spread). The models developed seek to simultaneously determine the size of the bid–ask spread and the time between trades. For the calls, the model is (7.55) (7.56) And symmetrically for the puts: (7.57) (7.58) where CBAi and PBAi are the call bid–ask spread and the put bid–ask spread for option i, respectively Ci and Pi are the call price and put price for option i, respectively CLi and PLi are the times between trades for the call and put option i, respectively CDUMi and PDUMi are dummy variables to allow for the minimum tick size

T is the time to maturity T2 allows for a non-linear relationship between time to maturity and the spread M2 is the square of moneyness, which is employed in quadratic form since at-the-money options have a higher trading volume, while out-of-the-money and in-the-money options both have lower trading activity 407

CRi and PRi are measures of risk for the call and put options, respectively, given by the square of their deltas. Equations (7.55) and (7.56), and then separately (7.57) and (7.58), are estimated using 2SLS. The results are given here in Tables 7.1 and 7.2. Table 7.1 Call bid–ask spread and trading volume regression

Note: t-ratios in parentheses. Source: George and Longstaff (1993). Reprinted with the permission of School of Business Administration, University of Washington.

Table 7.2 Put bid–ask spread and trading volume regression

Note: t-ratios in parentheses. Source: George and Longstaff (1993). Reprinted with the permission of School of Business Administration, University of Washington.

The adjusted R2 ≈ 0.6 for all four equations, indicating that the variables selected do a good job of explaining the spread and the time between trades. George and Longstaff (1993) argue that strategic market maker behaviour, which cannot be easily modelled, is important in influencing 408

the spread and that this precludes a higher adjusted R2. A next step in examining the empirical plausibility of the estimates is to consider the sizes, signs and significances of the coefficients. In the call and put spread regressions, respectively, α1 and β1 measure the tick size constraint on the spread – both are statistically significant and positive. α2 and β2 measure the effect of the option price on the spread. As expected, both of these coefficients are again significant and positive since these are inventory or holding costs. The coefficient value of approximately 0.017 implies that a one dollar increase in the price of the option will on average lead to a 1.7 cent increase in the spread. α3 and β3 measure the effect of trading activity on the spread. Recalling that an inverse trading activity variable is used in the regressions, again, the coefficients have their correct sign. That is, as the time between trades increases (that is, as trading activity falls), the bid–ask spread widens. Furthermore, although the coefficient values are small, they are statistically significant. In the put spread regression, for example, the coefficient of approximately 0.009 implies that, even if the time between trades widened from one minute to one hour, the spread would increase by only 54 cents. α4 and β4 measure the effect of time to maturity on the spread; both are negative and statistically significant. The authors argue that this may arise as market making is a more risky activity for near-maturity options. A possible alternative explanation, which they dismiss after further investigation, is that the early exercise possibility becomes more likely for very short-dated options since the loss of time value would be negligible. Finally, α5 and β5 measure the effect of risk on the spread; in both the call and put spread regressions, these coefficients are negative and highly statistically significant. This seems an odd result, which the authors struggle to justify, for it seems to suggest that more risky options will command lower spreads. Turning attention now to the trading activity regressions, γ1 and δ1 measure the effect of the spread size on call and put trading activity, respectively. Both are positive and statistically significant, indicating that a rise in the spread will increase the time between trades. The coefficients are such that a one cent increase in the spread would lead to an increase in the average time between call and put trades of nearly half a minute. γ2 and δ2 give the effect of an increase in time to maturity, while γ3 and δ3 are coefficients attached to the square of time to maturity. For both the call and put regressions, the coefficient on the level of time to maturity is 409

negative and significant, while that on the square is positive and significant. As time to maturity increases, the squared term would dominate, and one could therefore conclude that the time between trades will show a U-shaped relationship with time to maturity. Finally, γ4 and δ4 give the effect of an increase in the square of moneyness (i.e., the effect of an option going deeper into the money or deeper out of the money) on the time between trades. For both the call and put regressions, the coefficients are statistically significant and positive, showing that as the option moves further from the money in either direction, the time between trades rises. This is consistent with the authors’ supposition that trade is most active in at-the-money options, and less active in both out-of-the-money and in-themoney options.

7.9.6 Conclusions The value of the bid–ask spread on S&P100 index options and the time between trades (a measure of market liquidity) can be usefully modelled in a simultaneous system with exogenous variables such as the options’ deltas, time to maturity, moneyness, etc. This study represents a nice example of the use of a simultaneous equations system, but, in this author’s view, it can be criticised on several grounds. First, there are no diagnostic tests performed. Second, clearly the equations are all overidentified, but it is not obvious how the overidentifying restrictions have been generated. Did they arise from consideration of financial theory? For example, why do the CL and PL equations not contain the CR and PR variables? Why do the CBA and PBA equations not contain moneyness or squared maturity variables? The authors could also have tested for endogeneity of CBA and CL. Finally, the wrong sign on the highly statistically significant squared deltas is puzzling.

7.10 Vector Autoregressive Models Vector autoregressive models (VARs) were popularised in econometrics by Sims (1980) as a natural generalisation of univariate autoregressive models discussed in Chapter 6. A VAR is a systems regression model (i.e., there is more than one dependent variable) that can be considered a kind of hybrid between the univariate time series models considered in Chapter 6 and the simultaneous equations models developed previously in this chapter. VARs have often been advocated as an alternative to large-scale simultaneous equations structural models. 410

The simplest case that can be entertained is a bivariate VAR, where there are only two variables, y1t and y2t, each of whose current values depend on different combinations of the previous k values of both variables, and error terms (7.59) (7.60) where uit is a white noise disturbance term with E(uit) = 0, (i = 1, 2). Although, for simplicity, we usually assume that E(u1t u2t) = 0 so that the disturbances are uncorrelated across equations, it is common and more realistic to allow them to be contemporaneously correlated, so Cov(u1t u2t = σ12). As should already be evident, an important feature of the VAR model is its flexibility and the ease of generalisation. For example, the model could be extended to encompass moving average errors, which would be a multivariate version of an ARMA model, known as a VARMA. Instead of having only two variables, y1t and y2t, the system could also be expanded to include g variables, y1t, y2t, y3t, …, ygt, each of which has an equation. Another useful facet of VAR models is the compactness with which the notation can be expressed. For example, consider the case from above where k = 1, so that each variable depends only upon the immediately previous values of y1t and y2t, plus an error term. This could be written as (7.61) (7.62) or (7.63) or even more compactly as (7.64)

411

In equation (7.64), there are g = 2 variables in the system. Extending the model to the case where there are k lags of each variable in each equation is also easily accomplished using this notation (7.65) The model could be further extended to the case where the model includes first difference terms and cointegrating relationships (a vector error correction model (VECM) – see Chapter 8).

7.10.1 Advantages of VAR Modelling VAR models have several advantages compared with univariate time series models or simultaneous equations structural models The researcher does not need to specify which variables are endogenous or exogenous – all are endogenous. This is a very important point, since a requirement for simultaneous equations structural models to be estimable is that all equations in the system are identified. Essentially, this requirement boils down to a condition that some variables are treated as exogenous and that the equations contain different RHS variables. Ideally, this restriction should arise naturally from financial or economic theory. However, in practice theory will be at best vague in its suggestions of which variables should be treated as exogenous. This leaves the researcher with a great deal of discretion concerning how to classify the variables. Since Hausman-type tests are often not employed in practice when they should be, the specification of certain variables as exogenous, required to form identifying restrictions, is likely in many cases to be invalid. Sims (1980) termed these identifying restrictions ‘incredible’. VAR estimation, on the other hand, requires no such restrictions to be imposed. VARs allow the value of a variable to depend on more than just its own lags or combinations of white noise terms, so VARs are more flexible than univariate AR models; the latter can be viewed as a restricted case of VAR models. VAR models can therefore offer a very rich structure, implying that they may be able to capture more features of the data. Provided that there are no contemporaneous terms on the RHS of the equations and that the disturbances are uncorrelated across equations, 412

it is possible to simply use OLS separately on each equation. This arises from the fact that all variables on the RHS are pre-determined – that is, at time t, they are known. This implies that there is no possibility for feedback from any of the LHS variables to any of the RHS variables. Pre-determined variables include all exogenous variables and lagged values of the endogenous variables. If the disturbances are correlated, the equations could be jointly estimated using maximum likelihood (discussed at length in Chapter 9) based on the joint density of all the disturbances uit assuming multivariate normality. The forecasts generated by VARs are often better than ‘traditional structural’ models. It has been argued in a number of articles (see, for example, Sims, 1980) that large-scale structural models performed badly in terms of their out-of-sample forecast accuracy. This could perhaps arise as a result of the ad hoc nature of the restrictions placed on the structural models to ensure identification discussed above. McNees (1986) shows that forecasts for some variables (e.g., the US unemployment rate and real gross national product (GNP), etc.) are produced more accurately using VARs than from several different structural specifications.

7.10.2 Problems with VARs VAR models of course also have drawbacks and limitations relative to other model classes: VARs are a-theoretical (as are ARMA models), since they use little theoretical information about the relationships between the variables to guide the specification of the model. On the other hand, valid exclusion restrictions that ensure identification of equations from a simultaneous structural system will inform on the structure of the model. An upshot of this is that VARs are less amenable to theoretical analysis and therefore to policy prescriptions. There also exists an increased possibility under the VAR approach that a hapless researcher could obtain an essentially spurious relationship by mining the data. It is also often not clear how the VAR coefficient estimates should be interpreted. How should the appropriate lag lengths for the VAR be determined? There are several approaches available for dealing with this issue, which will be discussed below. 413

So many parameters! If there are g equations, one for each of g variables and with k lags of each of the variables in each equation, (g + kg2) parameters will have to be estimated. For example, if g = 3 and k = 3 there will be thirty parameters to estimate. For relatively small sample sizes, degrees of freedom will rapidly be used up, implying large standard errors and therefore wide confidence intervals for model coefficients. Should all of the components of the VAR be stationary? Obviously, if one wishes to use hypothesis tests, either singly or jointly, to examine the statistical significance of the coefficients, then it is essential that all of the components in the VAR are stationary. However, many proponents of the VAR approach recommend that differencing to induce stationarity should not be done. They would argue that the purpose of VAR estimation is purely to examine the relationships between the variables, and that differencing will throw information on any long-run relationships between the series away. It is also possible to combine levels and first differenced terms in a VECM – see Chapter 8.

7.10.3 Choosing the Optimal Lag Length for a VAR Often, financial theory will have little to say on what is an appropriate lag length for a VAR and how long changes in the variables should take to work through the system. In such instances, there are broadly three methods that could be used to arrive at the optimal lag length: rules of thumb, cross-equation restrictions and information criteria.

7.10.4 Rules of Thumb for VAR Lag Length Selection Similar to univariate AR(p) models, it might be possible to use the data frequency to decide the lag order and thus, for example, selecting p = 5 for daily data, p = 4 for quarterly data and so on. However, if the number of variables in the system is quite large, then a value of p this big, let alone the number that would by analogy be suggested for monthly data, would quickly become infeasible. It is also common to use an arbitrary fixed number of lags (typically, 1, 2, or 3) without further testing. Two more scientific approaches to choosing the lag order are given in the following sub-sections, but before moving on to those, it is worth noting that a high value of p may be required if the number of variables g in the system is too small and excludes relevant influences on the included 414

variables (possibly also including the wrong variables). Thus, in a sense, there is a trade-off between a larger p and a larger g. In such a situation, a better model is likely to arise from thinking creatively about additional variables to include in the model, even if these are hard to estimate, rather than increasing the lag length.

7.10.5 Cross-Equation Restrictions for VAR Lag Length Selection A first (but incorrect) response to the question of how to determine the appropriate lag length would be to use the block F-tests highlighted in Section 7.12 on page 319. These, however, are not appropriate in this case as the F-test would be used separately for the set of lags in each equation, and what is required here is a procedure to test the coefficients on a set of lags on all variables for all equations in the VAR at the same time. It is worth noting here that in the spirit of VAR estimation (as Sims, 1980, for example, thought that model specification should be conducted), the models should be as unrestricted as possible. A VAR with different lag lengths for each equation could be viewed as a restricted VAR. For example, consider a VAR with three lags of both variables in one equation and four lags of each variable in the other equation. This could be viewed as a restricted model where the coefficient on the fourth lags of each variable in the first equation have been set to zero. An alternative approach would be to specify the same number of lags in each equation and to determine the model order as follows. Suppose that a VAR estimated using quarterly data has eight lags of the two variables in each equation, and it is desired to examine a restriction that the coefficients on lags five–eight are jointly zero. This can be done using a likelihood ratio test (see Chapter 9 for more general details concerning such tests). Denote the variance–covariance matrix of residuals (given by ), as The likelihood ratio test for this joint hypothesis is given by (7.66) where is the determinant of the variance–covariance matrix of the residuals for the restricted model (with four lags), is the determinant of the variance–covariance matrix of residuals for the unrestricted VAR (with eight lags) and T is the sample size. The test statistic is asymptotically distributed as a χ2 variate with degrees of freedom equal to the total number of restrictions. In the VAR case above, four lags of two variables 415

are being restricted in each of the two equations = a total of 4 × 2 × 2 = 16 restrictions. In the general case of a VAR with g equations, to impose the restriction that the last q lags have zero coefficients, there would be g2q restrictions altogether. Intuitively, the test is a multivariate equivalent to examining the extent to which the RSS rises when a restriction is imposed. If and are ‘close together’, the restriction is supported by the data.

7.10.6 Information Criteria for VAR Lag Length Selection The likelihood ratio (LR) test explained above is intuitive and fairly easy to estimate, but has its limitations. Principally, one of the two VARs must be a special case of the other and, more seriously, only pairwise comparisons can be made. In the above example, if the most appropriate lag length had been seven or even ten, there is no way that this information could be gleaned from the LR test conducted. One could achieve this only by starting with a VAR(10), and successively testing one set of lags at a time. A further disadvantage of the LR test approach is that the χ2 test will strictly be valid asymptotically only under the assumption that the errors from each equation are normally distributed. This assumption is unlikely to be upheld for financial data. An alternative approach to selecting the appropriate VAR lag length would be to use an information criterion, as defined in Chapter 6 in the context of ARMA model selection. Information criteria require no such normality assumptions concerning the distributions of the errors. Instead, the criteria trade off a fall in the RSS of each equation as more lags are added, with an increase in the value of the penalty term. The univariate criteria could be applied separately to each equation but, again, it is usually deemed preferable to require the number of lags to be the same for each equation. This requires the use of multivariate versions of the information criteria, which can be defined as (7.67) (7.68) (7.69) where again is the variance–covariance matrix of residuals, T is the number of observations and k′ is the total number of regressors in all 416

equations, which will be equal to p2k + p for p equations in the VAR system, each with k lags of the p variables, plus a constant term in each equation. As previously, the values of the information criteria are constructed for lags (up to some pre-specified maximum ), and the chosen number of lags is that number minimising the value of the given information criterion.

7.11 Does the VAR Include Contemporaneous Terms? So far, it has been assumed that the VAR specified is of the form (7.70) (7.71) so that there are no contemporaneous terms on the RHS of equation (7.70) or (7.71) – i.e., there is no term in y2t on the RHS of the equation for y1t and no term in y1t on the RHS of the equation for y2t. But what if the equations had a contemporaneous feedback term, as in the following case? (7.72) (7.73) Equations (7.72) and (7.73) could also be written by stacking up the terms into matrices and vectors: (7.74) This would be known as a VAR in primitive form, similar to the structural form for a simultaneous equations model. Some researchers have argued that the a-theoretical nature of reduced form VARs leaves them unstructured and their results difficult to interpret theoretically. They argue that the forms of VAR given previously are merely reduced forms of a more general structural VAR (such as equation (7.74)), with the latter being of more interest. The contemporaneous terms from equation (7.74) can be taken over to the LHS and written as

417

(7.75) or (7.76) If both sides of equation (7.76) are pre-multiplied by A–1 (7.77) or (7.78) This is known as a standard form VAR, which is akin to the reduced form from a set of simultaneous equations. This VAR contains only predetermined values on the RHS (i.e., variables whose values are known at time t), and so there is no contemporaneous feedback term. This VAR can therefore be estimated equation by equation using OLS. Equation (7.74), the structural or primitive form VAR, is not identified, since identical pre-determined (lagged) variables appear on the RHS of both equations. In order to circumvent this problem, a restriction that one of the coefficients on the contemporaneous terms is zero must be imposed. In equation (7.74), either α12 or α22 must be set to zero to obtain a triangular set of VAR equations that can be validly estimated. The choice of which of these two restrictions to impose is ideally made on theoretical grounds. For example, if financial theory suggests that the current value of y1t should affect the current value of y2t but not the other way around, set α12 = 0, and so on. Another possibility would be to run separate estimations, first imposing α12 = 0 and then α22 = 0, to determine whether the general features of the results are much changed. It is also very common to estimate only a reduced form VAR, which is of course perfectly valid provided that such a formulation is not at odds with the relationships between variables that financial theory says should hold. One fundamental weakness of the VAR approach to modelling is that its a-theoretical nature and the large number of parameters involved make the estimated models difficult to interpret. In particular, some lagged variables may have coefficients which change sign across the lags, and this, together with the interconnectivity of the equations, could render it difficult to see 418

what effect a given change in a variable would have upon the future values of the variables in the system. In order to partially alleviate this problem, three sets of statistics are usually constructed for an estimated VAR model: block significance tests, impulse responses and variance decompositions. How important an intuitively interpretable model is will of course depend on the purpose of constructing the model. Interpretability may not be an issue at all if the purpose of producing the VAR is to make forecasts – see Box 7.3. BOX 7.3 Forecasting with VARs One of the main advantages of the VAR approach to modelling and forecasting is that since only lagged variables are used on the RHS, forecasts of the future values of the dependent variables can be calculated iteratively using only information from within the system, very similar in approach to the manner in which forecasts are derived from an AR(p) model. We could term these unconditional forecasts since they are not constructed conditional on a particular set of assumed values. However, conversely it may be useful to produce forecasts of the future values of some variables conditional upon known values of other variables in the system. For example, it may be the case that the values of some variables become known before the values of the others. If the known values of the former are employed, we would anticipate that the forecasts should be more accurate than if estimated values were used unnecessarily, thus throwing known information away. Alternatively, conditional forecasts can be employed for counterfactual analysis based on examining the impact of certain scenarios. For example, given a trivariate VAR system incorporating monthly stock returns, inflation and gross domestic product (GDP), we could answer the question: ‘What is the likely impact on the stock market over the next 1–6 months of a 2-percentage point increase in inflation and a 1% rise in GDP?’ Usually, the forecasts from a VAR are evaluated separately equation-by-equation and then the forecasts can be compared with those of other approaches (such as a linear regression or AR(p) model) using the standard forecast error aggregation tools such as RMSE as discussed in Chapter 6. Indeed, it will often be the case that only the forecasts from one equation in the VAR are actually of interest.

419

7.12 Block Significance and Causality Tests It is likely that, when a VAR includes many lags of variables, it will be difficult to see which sets of variables have significant effects on each dependent variable and which do not. In order to address this issue, tests are usually conducted that restrict all of the lags of a particular variable to zero. For illustration, consider the following bivariate VAR(3)

(7.79)

This VAR could be written out to express the individual equations as

(7.80)

One might be interested in testing the hypotheses and their implied restrictions on the parameter matrices given in Table 7.3. Table 7.3 Granger causality tests and implied restrictions on VAR models Hypothesis

Implied restriction

1 Lags of y1t do not explain current y2t

β21 = 0 and γ21 = 0 and δ21 = 0

2 Lags of y1t do not explain current y1t

β11 = 0 and γ11 = 0 and δ11 = 0

3 Lags of y2t do not explain current y1t

β12 = 0 and γ12 = 0 and δ12 = 0

4 Lags of y2t do not explain current y2t

β22 = 0 and γ22 = 0 and δ22 = 0

Assuming that all of the variables in the VAR are stationary, the joint 420

hypotheses can easily be tested within the F-test framework, since each individual set of restrictions involves parameters drawn from only one equation. The equations would be estimated separately using OLS to obtain the unrestricted RSS, then the restrictions imposed and the models re-estimated to obtain the restricted RSS. The F-statistic would then take the usual form described in Chapter 4. Thus, evaluation of the significance of variables in the context of a VAR almost invariably occurs on the basis of joint tests on all of the lags of a particular variable in an equation, rather than by examination of individual coefficient estimates. In fact, the tests described above could also be referred to as causality tests. Tests of this form were described by Granger (1969) and a slight variant due to Sims (1972). Causality tests seek to answer simple questions of the type, ‘Do changes in y1 cause changes in y2?’ The argument follows that if y1 causes y2, lags of y1 should be significant in the equation for y2. If this is the case and not vice versa, it would be said that y1 ‘Grangercauses’ y2 or that there exists unidirectional causality from y1 to y2. On the other hand, if y2 causes y1, lags of y2 should be significant in the equation for y1. If both sets of lags were significant, it would be said that there was ‘bi-directional causality’ or ‘bi-directional feedback’. If y1 is found to Granger-cause y2, but not vice versa, it would be said that variable y1 is strongly exogenous (in the equation for y2). If neither set of lags are statistically significant in the equation for the other variable, it would be said that y1 and y2 are independent. Finally, the word ‘causality’ is somewhat of a misnomer, for Granger-causality really means only a correlation between the current value of one variable and the past values of others; it does not mean that movements of one variable cause movements of another.

7.12.1 Restricted VARs As written above in, for example, equations (7.59) and (7.60), the VAR is completely unrestricted and as general as possible in the sense that the lags of every variable in the system enter into the equations for every variable. However, as discussed this can lead to very highly parameterised systems with surprisingly few degrees of freedom even when the number of timeseries observations is quite large. Such systems may also produce relatively poor forecasts as a result. It is possible that theory might suggest that certain lagged variables 421

should not appear in certain equations, in which case a restricted or unbalanced VAR with fewer parameters could be formed. For example, if we were measuring interactions between small and large economies, it might be plausible to assume that whatever goes on in the small economy cannot affect the large one and thus we might set the lags of the former to zero in the equation(s) for the latter. Alternatively, it might be that the block significance tests outlined in the previous section suggested that particular sets of lags could be removed from one or more equations. However, it probably does not make sense to remove specific lags of individual variables from an equation while leaving other lags of the same variable in the equation (e.g., removing y2t–1 and y2t–3 from the equation for y1t but leaving y2t–2): either a given variable influences another variable or it does not. An alternative way to reduce the number of parameters to estimate in a VAR is to use a Bayesian approach, where specific prior distributions are imposed on the VAR to leave fewer free parameters than originally. See Doan et al. (1984) or Giannone, Lenza and Primiceri (2014) and the references therein for further details.

7.13 VARs with Exogenous Variables Consider the following specification for a VAR(1) where Xt is a vector of exogenous variables and B is a matrix of coefficients (7.81) The components of the vector Xt are known as exogenous variables since their values are determined outside of the VAR system – in other words, there are no equations in the VAR with any of the components of Xt as dependent variables. Such a model is sometimes termed a VARX, although it could be viewed as simply a very restricted VAR where there are equations for each of the exogenous variables, but with the coefficients on the RHS in those equations restricted to zero. Such a restriction may be considered desirable if theoretical considerations suggest it, although it is clearly not in the true spirit of VAR modelling, which is not to impose any restrictions on the model but rather to ‘let the data decide’.

7.14 Impulse Responses and Variance Decompositions 422

Block F-tests and an examination of causality in a VAR will suggest which of the variables in the model have statistically significant impacts on the future values of each of the variables in the system. But F-test results will not, by construction, be able to explain the sign of the relationship or how long these effects require to take place. That is, F-test results will not reveal whether changes in the value of a given variable have a positive or negative effect on other variables in the system, or how long it would take for the effect of that variable to work through the system. Such information will, however, be given by an examination of the VAR’s impulse responses and variance decompositions. Impulse responses trace out the responsiveness of the dependent variables in the VAR to shocks to each of the variables. So, for each variable from each equation separately, a unit shock is applied to the error, and the effects upon the VAR system over time are noted. Effectively, the impulse responses are partial derivatives of the variables (yjt, j = 1, …, g) with respect to each error term In practice, one standard deviation shocks are often used rather than one unit, as it might be the case that a one unit shock is empirically implausible, but a one standard deviation shock will almost always be relevant. If there are g variables in a system, a total of g2 impulse responses could be generated. The way that this is achieved in practice is by expressing the VAR model as a VMA – that is, the vector autoregressive model is written as a vector moving average (in the same way as was done for univariate autoregressive models in Chapter 5). Provided that the system is stable, the shock should gradually die away. To illustrate how impulse responses operate, consider the following bivariate VAR(1)

(7.82)

The VAR can also be written out using the elements of the matrices and vectors as (7.83)

423

Consider the effect at time t = 0, 1, …, of a unit shock to y1t at time t = 0, where the T × 1 vector for y at time t is stacked up and written as simply [yt ] for notational convenience (7.84) (7.85) (7.86) and so on. It would thus be possible to plot the impulse response functions of y1t and y2t to a unit shock in y1t. Notice that the effect on y2t is always zero, since the variable y1t–1 has a zero coefficient attached to it in the equation for y2t. Now consider the effect of a unit shock to y2t at time t = 0 (7.87) (7.88) (7.89) and so on. Although it is probably fairly easy to see what the effects of shocks to the variables will be in such a simple VAR, the same principles can be applied in the context of VARs containing more equations or more lags, where it is much more difficult to see by eye what are the interactions between the equations. Variance decompositions offer a slightly different method for examining VAR system dynamics. They give the proportion of the movements in the dependent variables that are due to their ‘own’ shocks, versus shocks to the other variables. A shock to the ith variable will directly affect that variable of course, but it will also be transmitted to all of the other variables in the system through the dynamic structure of the VAR. Variance decompositions determine how much of the s-step-ahead forecast error variance of a given variable is explained by innovations to each 424

explanatory variable for s = 1, 2, … In practice, it is usually observed that own series shocks explain most of the (forecast) error variance of the series in a VAR. To some extent, impulse responses and variance decompositions offer very similar information. For calculating impulse responses and variance decompositions, the ordering of the variables is important. To see why this is the case, recall that the impulse responses refer to a unit shock to the errors of one VAR equation alone. This implies that the error terms of all other equations in the VAR system are held constant. However, this is not realistic since the error terms are likely to be correlated across equations to some extent. Thus, assuming that they are completely independent would lead to a misrepresentation of the system dynamics. In practice, the errors will have a common component that cannot be associated with a single variable alone. The usual approach to this difficulty is to generate orthogonalised impulse responses. In the context of a bivariate VAR, the whole of the common component of the errors is attributed somewhat arbitrarily to the first variable in the VAR. In the general case where there are more than two variables in the VAR, the calculations are more complex but the interpretation is the same. Such a restriction in effect implies an ‘ordering’ of variables, so that the equation for y1t would be estimated first and then that of y2t, a bit like a recursive or triangular system. Assuming a particular ordering is necessary to compute the impulse responses and variance decompositions. Ideally, financial theory should suggest an ordering (in other words, that movements in some variables are likely to follow, rather than precede, others). Failing this, the sensitivity of the results to changes in the ordering can be observed by assuming one ordering, and then exactly reversing it and recomputing the impulse responses and variance decompositions. It is also worth noting that the more highly correlated are the residuals from an estimated equation, the more the variable ordering will be important. But when the residuals are almost uncorrelated, the ordering of the variables will make little difference (see Lütkepohl, 1991, Chapter 2 for further details). Runkle (1987) argues that both impulse responses and variance decompositions are notoriously difficult to interpret accurately. He argues that confidence bands around the impulse responses and variance decompositions should always be constructed. However, he further states that, even then, the confidence intervals are typically so wide that sharp inferences are impossible. 425

7.15 VAR Model Example: The Interaction Between Property Returns and the Macroeconomy 7.15.1 Background, Data and Variables Brooks and Tsolacos (1999) employ a VAR methodology for investigating the interaction between the UK property market and various macroeconomic variables. Monthly data, in logarithmic form, are used for the period from December 1985 to January 1998. The selection of the variables for inclusion in the VAR model is governed by the time series that are commonly included in studies of stock return predictability. It is assumed that stock returns are related to macroeconomic and business conditions, and hence time series which may be able to capture both current and future directions in the broad economy and the business environment are used in the investigation. Broadly, there are two ways to measure the value of property-based assets – direct measures of property value and equity-based measures. Direct property measures are based on periodic appraisals or valuations of the actual properties in a portfolio by surveyors, while equity-based measures evaluate the worth of properties indirectly by considering the values of stock market traded property companies. Both sources of data have their drawbacks. Appraisal-based value measures suffer from valuation biases and inaccuracies. Surveyors are typically prone to ‘smooth’ valuations over time, such that the measured returns are too low during property market booms and too high during periods of property price falls. Additionally, not every property in the portfolio that comprises the value measure is appraised during every period, resulting in some stale valuations entering the aggregate valuation, further increasing the degree of excess smoothness of the recorded property price series. Indirect property vehicles – property-related companies traded on stock exchanges – do not suffer from the above problems, but are excessively influenced by general stock market movements. It has been argued, for example, that over three-quarters of the variation over time in the value of stock exchange traded property companies can be attributed to general stock market-wide price movements. Therefore, the value of equity-based property series reflects much more the sentiment in the general stock market than the sentiment in the property market specifically. Brooks and Tsolacos (1999) elect to use the equity-based FTSE Property Total Return Index to construct property returns. In order to purge the real estate return series of its general stock market influences, it 426

is common to regress property returns on a general stock market index (in this case the FTA All-Share Index is used), saving the residuals. These residuals are expected to reflect only the variation in property returns, and thus become the property market return measure used in subsequent analysis, and are denoted PROPRES. Hence, the variables included in the VAR are the property returns (with general stock market effects removed), the rate of unemployment, nominal interest rates, the spread between the long- and short-term interest rates, unanticipated inflation and the dividend yield. The motivations for including these particular variables in the VAR together with the property series, are as follows The rate of unemployment (denoted UNEM) is included to indicate general economic conditions. In US research, authors tend to use aggregate consumption, a variable that has been built into asset pricing models and examined as a determinant of stock returns. Data for this variable and for alternative variables such as GDP are not available on a monthly basis in the UK. Monthly data are available for industrial production series but other studies have not shown any evidence that industrial production affects real estate returns. As a result, this series was not considered as a potential causal variable. Short-term nominal interest rates (denoted SIR) are assumed to contain information about future economic conditions and to capture the state of investment opportunities. It was found in previous studies that short-term interest rates have a very significant negative influence on property stock returns. Interest rate spreads (denoted SPREAD), i.e., the yield curve, are usually measured as the difference in the returns between long-term Treasury Bonds (of maturity, say, ten or twenty years), and the onemonth or three-month Treasury Bill rate. It has been argued that the yield curve has extra predictive power, beyond that contained in the short-term interest rate, and can help predict GDP up to four years ahead. It has also been suggested that the term structure also affects real estate market returns. Inflation rate influences are also considered important in the pricing of stocks. For example, it has been argued that unanticipated inflation could be a source of economic risk and as a result, a risk premium will also be added if the stock of firms has exposure to unanticipated inflation. The unanticipated inflation variable (denoted UNINFL) is defined as the difference between the realised inflation rate, computed 427

as the percentage change in the Retail Price Index (RPI), and an estimated series of expected inflation. The latter series was produced by fitting an ARMA model to the actual series and making a oneperiod(month)-ahead forecast, then rolling the sample forward one period, and re-estimating the parameters and making another onestep-ahead forecast, and so on. Dividend yields (denoted DIVY) have been widely used to model stock market returns, and also real estate property returns, based on the assumption that movements in the dividend yield series are related to long-term business conditions and that they capture some predictable components of returns. All variables to be included in the VAR are required to be stationary in order to carry out joint significance tests on the lags of the variables. Hence, all variables are subjected to augmented Dickey–Fuller (ADF) tests (see Chapter 8). Evidence that the log of the RPI and the log of the unemployment rate both contain a unit root is observed. Therefore, the first differences of these variables are used in subsequent analysis. The remaining four variables led to rejection of the null hypothesis of a unit root in the log-levels, and hence these variables were not first differenced.

7.15.2 Methodology A reduced form VAR is employed and therefore each equation can effectively be estimated using OLS. For a VAR to be unrestricted, it is required that the same number of lags of all of the variables is used in all equations. Therefore, in order to determine the appropriate lag lengths, the multivariate generalisation of Akaike’s information criterion (AIC) is used. Within the framework of the VAR system of equations, the significance of all the lags of each of the individual variables is examined jointly with an F-test. Since several lags of the variables are included in each of the equations of the system, the coefficients on individual lags may not appear significant for all lags, and may have signs and degrees of significance that vary with the lag length. However, F-tests will be able to establish whether all of the lags of a particular variable are jointly significant. In order to consider further the effect of the macro-economy on the real estate returns index, the impact multipliers (orthogonalised impulse responses) are also calculated for the estimated VAR model. Two standard error bands are calculated using the Monte Carlo integration approach employed by McCue and Kling (1994), and based on Doan (1984). The forecast error 428

variance is also decomposed to determine the proportion of the movements in the real estate series that are a consequence of its own shocks rather than shocks to other variables.

7.15.3 Results The number of lags that minimises the value of Akaike’s information criterion is fourteen, consistent with the fifteen lags used by McCue and Kling (1994). There are thus (1 + 14 × 6) = 85 variables in each equation, implying fifty-nine degrees of freedom. F-tests for the null hypothesis that all of the lags of a given variable are jointly insignificant in a given equation are presented in Table 7.4. Table 7.4 Marginal significance levels associated with joint F-tests Lags of variable variable

SIR

SIR

DIVY

SPREAD

UNEM

UNINFL

PROPRES

0.0000 0.0091 0.0242

0.0327

0.2126

0.0000

DIVY

0.5025 0.0000 0.6212

0.4217

0.5654

0.4033

SPREAD

0.2779 0.1328 0.0000

0.4372

0.6563

0.0007

UNEM

0.3410 0.3026 0.1151

0.0000

0.0758

0.2765

UNINFL

0.3057 0.5146 0.3420

0.4793

0.0004

0.3885

PROPRES

0.5537 0.1614 0.5537

0.8922

0.7222

0.0000

The test is that all fourteen lags have no explanatory power for that particular equation in the VAR. Source: Brooks and Tsolacos (1999).

In contrast to a number of US studies which have used similar variables, it is found to be difficult to explain the variation in the UK real estate returns index using macroeconomic factors, as the last row of Table 7.4 shows. Of all the lagged variables in the real estate equation, only the lags of the real estate returns themselves are highly significant, and the dividend yield variable is significant only at the 20% level. No other variables have any significant explanatory power for the real estate returns. Therefore, based on the F-tests, an initial conclusion is that the variation in property returns, net of stock market influences, cannot be explained by any of the main macroeconomic or financial variables used in existing 429

research. One possible explanation for this might be that, in the UK, these variables do not convey the information about the macro-economy and business conditions assumed to determine the intertemporal behaviour of property returns. It is possible that property returns may reflect property market influences, such as rents, yields or capitalisation rates, rather than macroeconomic or financial variables. However, again the use of monthly data limits the set of both macroeconomic and property market variables that can be used in the quantitative analysis of real estate returns in the UK. It appears, however, that lagged values of the real estate variable have explanatory power for some other variables in the system. These results are shown in the last column of Table 7.4. The property sector appears to help in explaining variations in the term structure and short-term interest rates, and moreover since these variables are not significant in the property index equation, it is possible to state further that the property residual series Granger-causes the short-term interest rate and the term spread. This is a bizarre result. The fact that property returns are explained by own lagged values – i.e., that is there is interdependency between neighbouring data points (observations) – may reflect the way that property market information is produced and reflected in the property return indices. Table 7.5 gives variance decompositions for the property returns index equation of the VAR for one, two, three, four, twelve and twenty-four steps ahead for the two variable orderings Order I: PROPRES, DIVY, UNINFL, UNEM, SPREAD, SIR Order II: SIR, SPREAD, UNEM, UNINFL, DIVY, PROPRES Unfortunately, the ordering of the variables is important in the decomposition. Thus two orderings are applied, which are the exact opposite of one another, and the sensitivity of the result is considered. It is clear that by the two-year forecasting horizon, the variable ordering has become almost irrelevant in most cases. An interesting feature of the results is that shocks to the term spread and unexpected inflation together account for over 50% of the variation in the real estate series. The shortterm interest rate and dividend yield shocks account for only 10–15% of the variance of the property index. One possible explanation for the difference in results between the F-tests and the variance decomposition is that the former is a causality test and the latter is effectively an exogeneity test. Hence the latter implies the stronger restriction that both current and lagged shocks to the explanatory variables do not influence the current 430

value of the dependent variable of the property equation. Another way of stating this is that the term structure and unexpected inflation have a contemporaneous rather than a lagged effect on the property index, which implies insignificant F-test statistics but explanatory power in the variance decomposition. Therefore, although the F-tests did not establish any significant effects, the error variance decompositions show evidence of a contemporaneous relationship between PROPRES and both SPREAD and UNINFL. The lack of lagged effects could be taken to imply speedy adjustment of the market to changes in these variables. Table 7.5 Variance decompositions for the property sector index residuals Explained by innovations in Months ahead

SIR I

DIVY II

I

II

SPREAD

UNEM

UNINFL

I

II

I

I

II

II

1

0.0 0.8 0.0

38.2 0.0

9.1

0.0 0.7

0.0

0.2

2

0.2 0.8 0.2

35.1 0.2

12.3 0.4 1.4

1.6

2.9

3

3.8 2.5 0.4

29.4 0.2

17.8 1.0 1.5

2.3

3.0

4

3.7 2.1 5.3

22.3 1.4

18.5 1.6 1.1

4.8

4.4

12

2.8 3.1 15.5 8.7

15.3 19.5 3.3 5.1

24

8.2 6.3 6.8

38.0 36.2 5.5 14.7 18.1 16.9

3.9

17.0 13.5

Source: Brooks and Tsolacos (1999).

Figures 7.1 and 7.2 give the impulse responses for PROPRES associated with separate unit shocks to unexpected inflation and the dividend yield, as examples (as stated above, a total of thirty-six impulse responses could be calculated since there are six variables in the system).

431

Figure 7.1 Impulse responses and standard error bands for innovations in

unexpected inflation equation errors

Figure 7.2 Impulse responses and standard error bands for innovations in

the dividend yields Considering the signs of the responses, innovations to unexpected inflation (Figure 7.1) always have a negative impact on the real estate index, since the impulse response is negative, and the effect of the shock does not die down, even after twenty-four months. Increasing stock dividend yields (Figure 7.2) have a negative impact for the first three periods, but beyond that, the shock appears to have worked its way out of the system.

7.15.4 Conclusions The conclusion from the VAR methodology adopted in the Brooks and Tsolacos paper is that overall, UK real estate returns are difficult to explain on the basis of the information contained in the set of the variables 432

used in existing studies based on non-UK data. The results are not strongly suggestive of any significant influences of these variables on the variation of the filtered property returns series. There is, however, some evidence that the interest rate term structure and unexpected inflation have a contemporaneous effect on property returns, in agreement with the results of a number of previous studies.

7.16 A Couple of Final Points on VARs The VAR approach to model-building has become enormously popular over the past three decades, due in part to their simplicity but also to their flexibility. Two further ways in which the standard approach to VAR construction can be extended are The possibility of latent (hidden) variables can be accounted for by using a VAR in state space form, which can then be estimated via the Kalman filter (see Chapter 15 of this book for further details on the latter) Non-linear VARs can be constructed, incorporating Markov switching regimes or threshold dynamics (see Chapter 10 for a discussion of these models) The free e-book by Ouliaris, Pagan and Restrepo (2016) contains further details on many aspects of VAR models and their implementation in EViews. KEY CONCEPTS The key terms to be able to define and explain from this chapter are endogenous variable simultaneous equations bias order condition Hausman test structural form indirect least squares vector autoregression impulse response exogenous variable identified 433

rank condition reduced form instrumental variables two-stage least squares Granger causality variance decomposition

SELF-STUDY QUESTIONS 1. Consider the following simultaneous equations system (7.90) (7.91) (7.92) (a) Derive the reduced form equations corresponding to equations (7.90)–(7.92). (b) What do you understand by the term ‘identification’? Describe a rule for determining whether a system of equations is identified. Apply this rule to equations (7.90)–(7.92). Does this rule guarantee that estimates of the structural parameters can be obtained? (c) Which would you consider the more serious misspecification: treating exogenous variables as endogenous, or treating endogenous variables as exogenous? Explain your answer. (d) Describe a method of obtaining the structural form coefficients corresponding to an overidentified system. (e) Using EViews, estimate a VAR model for the interest rate series used in the principal components example of Chapter 4. Use a method for selecting the lag length in the VAR optimally. Determine whether certain maturities lead or lag others, by conducting Granger causality tests and plotting impulse responses and variance decompositions. Is there any evidence that new information is reflected more quickly in 434

some maturities than others? 2. Consider the following system of two equations (7.93) (7.94) (a) Explain, with reference to these equations, the undesirable consequences that would arise if equations (7.93) and (7.94) were estimated separately using OLS. (b) What would be the effect upon your answer to (a) if the variable y1t had not appeared in equation (7.94)? (c) State the order condition for determining whether an equation which is part of a system is identified. Use this condition to determine whether equations (7.93) or (7.94) or both or neither are identified. (d) Explain whether indirect least squares (ILS) or two-stage least squares (2SLS) could be used to obtain the parameters of equation (7.93) and (7.94). Describe how each of these two procedures (ILS and 2SLS) are used to calculate the parameters of an equation. Compare and evaluate the usefulness of ILS, 2SLS and IV. (e) Explain briefly the Hausman procedure for testing for exogeneity. 3. Explain, using an example if you consider it appropriate, what you understand by the equivalent terms ‘recursive equations’ and ‘triangular system’. Can a triangular system be validly estimated using OLS? Explain your answer. 4. Consider the following vector autoregressive model (7.95) where yt is a p × 1 vector of variables determined by k lags of all p variables in the system, ut is a p× 1 vector of error terms, β0 is a p× 1 vector of constant term coefficients and βi are p × p matrices of coefficients on the ith lag of y. 435

(a) If p = 2, and k = 3, write out all the equations of the VAR in full, carefully defining any new notation you use that is not given in the question. (b) Why have VARs become popular for application in economics and finance, relative to structural models derived from some underlying theory? (c) Discuss any weaknesses you perceive in the VAR approach to econometric modelling. (d) Two researchers, using the same set of data but working independently, arrive at different lag lengths for the VAR equation (7.95). Describe and evaluate two methods for determining which of the lag lengths is more appropriate. 5. Define carefully the following terms Simultaneous equations system Exogenous variables Endogenous variables Structural form model Reduced form model.

436

8 Modelling Long-Run Relationships in Finance

LEARNING OUTCOMES In this chapter, you will learn how to Highlight the problems that may occur if non-stationary data are used in their levels form Test for unit roots Examine whether systems of variables are cointegrated Estimate error correction and vector error correction models Explain the intuition behind Johansen’s test for cointegration Describe how to test hypotheses in the Johansen framework

8.1 Stationarity and Unit Root Testing 8.1.1 Why are Tests for Non-Stationarity Necessary? There are several reasons why the concept of non-stationarity is important and why it is essential that variables that are non-stationary be treated differently from those that are stationary. Two definitions of nonstationarity were presented at the start of Chapter 6. For the purpose of the analysis in this chapter, a stationary series can be defined as one with a constant mean, constant variance and constant autocovariances for each given lag. Therefore, the discussion in this chapter relates to the concept of weak stationarity. An examination of whether a series can be viewed as stationary or not is essential for the following reasons The stationarity or otherwise of a series can strongly influence its 437

behaviour and properties. To offer one illustration, the word ‘shock’ is usually used to denote a change or an unexpected change in a variable or perhaps simply the value of the error term during a particular time period. For a stationary series, ‘shocks’ to the system will gradually die away. That is, a shock during time t will have a smaller effect in time t + 1, a smaller effect still in time t + 2, and so on. This can be contrasted with the case of non-stationary data, where the persistence of shocks will always be infinite, so that for a nonstationary series, the effect of a shock during time t will not have a smaller effect in time t +1, and in time t +2, etc. The use of non-stationary data can lead to spurious regressions. If two stationary variables are generated as independent random series, when one of those variables is regressed on the other, the t-ratio on the slope coefficient would be expected not to be significantly different from zero, and the value of R2 would be expected to be very low. This seems obvious, for the variables are not related to one another. However, if two variables are trending over time, a regression of one on the other could have a high R2 even if the two are totally unrelated. So, if standard regression techniques are applied to non-stationary data, the end result could be a regression that ‘looks’ good under standard measures (significant coefficient estimates and a high R2), but which is really valueless. Such a model would be termed a ‘spurious regression’. To give an illustration of this, two independent sets of nonstationary variables, y and x, were generated with sample size 500, one regressed on the other and the R2 noted. This was repeated 1,000 times to obtain 1,000 R2 values. A histogram of these values is given in Figure 8.1. As Figure 8.1 shows, although one would have expected the R2 values for each regression to be close to zero, since the explained and explanatory variables in each case are independent of one another, in fact R2 takes on values across the whole range. For one set of data, R2 is bigger than 0.9, while it is bigger than 0.5 over 16% of the time! If the variables employed in a regression model are not stationary, then it can be proved that the standard assumptions for asymptotic analysis will not be valid. In other words, the usual ‘t-ratios’ will not follow a t-distribution, and the F-statistic will not follow an Fdistribution, and so on. Using the same simulated data as used to produce Figure 8.1, Figure 8.2 plots a histogram of the estimated t438

ratio on the slope coefficient for each set of data. In general, if one variable is regressed on another unrelated variable, the t-ratio on the slope coefficient will follow a tdistribution. For a sample of size 500, this implies that 95% of the time, the t-ratio will lie between ±2. As Figure 8.2 shows quite dramatically, however, the standard t-ratio in a regression of nonstationary variables can take on enormously large values. In fact, in the above example, the t-ratio is bigger than 2 in absolute value over 98% of the time, when it should be bigger than 2 in absolute value only approximately 5% of the time! Clearly, it is therefore not possible to validly undertake hypothesis tests about the regression parameters if the data are non-stationary.

Figure 8.1 Value of R2 for 1000 sets of regressions of a non-stationary

variable on another independent non-stationary variable

439

Figure 8.2 Value of t-ratio of slope coefficient for 1,000 sets of

regressions of a non-stationary variable on another independent nonstationary variable

8.1.2 Two Types of Non-Stationarity There are two models that have been frequently used to characterise the non-stationarity, the random walk model with drift (8.1) and the trend-stationary process – so called because it is stationary around a linear trend (8.2) where ut is a white noise disturbance term in both cases. Note that the model (8.1) could be generalised to the case where yt is an explosive process (7.47) where ϕ > 1. Typically, this case is ignored and ϕ = 1 is used to characterise the non-stationarity because ϕ > 1 does not describe many data series in economics and finance, but ϕ = 1 has been found to describe accurately many financial and economic time series. Moreover, ϕ > 1 has an intuitively unappealing property: shocks to the system are not only persistent through time, they are propagated so that a given shock will 440

have an increasingly large influence. In other words, the effect of a shock during time t will have a larger effect in time t + 1, a larger effect still in time t + 2, and so on. To see this, consider the general case of an AR(1) with no drift (8.4) Let ϕ take any value for now. Lagging equation (8.4) one and then two periods (8.5) (8.6) Substituting into equation (8.4) from equation (8.5) for yt−1 yields (8.7) (8.8) Substituting again for yt−2 from equation (8.6) (8.9) (8.10) T successive substitutions of this type lead to (8.11) There are three possible cases: (1) ϕ < 1 ⇒ ϕT → 0 as T → ∞ So the shocks to the system gradually die away – this is the stationary case. (2) ϕ = 1 ⇒ ϕT = 1 ∀ T So shocks persist in the system and never die away. The following is obtained (8.12) So the current value of y is just an infinite sum of past shocks plus 441

some starting value of y0. This is known as the unit root case, for the root of the characteristic equation would be unity. (3) ϕ > 1. Now given shocks become more influential as time goes on, since if ϕ > 1, ϕ3 > ϕ2 > ϕ, etc. This is the explosive case which, for the reasons listed above, will not be considered as a plausible description of the data. Going back to the two characterisations of non-stationarity, the random walk with drift (8.13) and the trend-stationary process (8.14) The two will require different treatments to induce stationarity. The second case is known as deterministic non-stationarity and de-trending is required. In other words, if it is believed that only this class of nonstationarity is present, a regression of the form given in equation (8.14) would be run, and any subsequent estimation would be done on the residuals from equation (8.14), which would have had the linear trend removed. The first case is known as stochastic non-stationarity, where there is a stochastic trend in the data. Letting Δyt = yt − yt−1 and Lyt = yt−1 so that (1 − L) yt = yt − Lyt = yt − yt−1. If equation (8.13) is taken and yt−1 subtracted from both sides (8.15) (8.16) (8.17) There now exists a new variable Δyt, which will be stationary. It would be said that stationarity has been induced by ‘differencing once’. It should also be apparent from the representation given by equation (8.16) why yt is also known as a unit root process: i.e., that the root of the characteristic equation (1− z) = 0, will be unity. Although trend-stationary and difference-stationary series are both 442

‘trending’ over time, the correct approach needs to be used in each case. If first differences of a trend-stationary series were taken, it would ‘remove’ the non-stationarity, but at the expense of introducing an MA(1) structure into the errors. To see this, consider the trend-stationary model (8.18) This model can be expressed for time t − 1, which would be obtained by removing 1 from all of the time subscripts in equation (8.18) (8.19) Subtracting equation (8.19) from equation (8.18) gives (8.20) Not only is this a moving average in the errors that has been created, it is a noninvertible MA (i.e., one that cannot be expressed as an autoregressive process). Thus the series, Δyt would in this case have some very undesirable properties. Conversely if one tried to de-trend a series which has stochastic trend, then the non-stationarity would not be removed. Clearly then, it is not always obvious which way to proceed. One possibility is to nest both cases in a more general model and to test that. For example, consider the model (8.21) Although again, of course the t-ratios in equation (8.21) will not follow a tdistribution and thus hypotheses about these parameters cannot be tested unless y is actually stationary in levels. Such a model could allow for both deterministic and stochastic non-stationarity. However, this book will now concentrate on the stochastic stationarity model since it is the model that has been found to best describe most non-stationary financial and economic time series. Consider again the simplest stochastic trend model (8.22) or (8.23) 443

This concept can be generalised to consider the case where the series contains more than one ‘unit root’. That is, the first difference operator, Δ, would need to be applied more than once to induce stationarity. This situation will be described later in this chapter. Arguably the best way to understand the ideas discussed above is to consider some diagrams showing the typical properties of certain relevant types of processes. Figure 8.3 plots a white noise (pure random) process, while Figures 8.4 and 8.5 plot a random walk versus a random walk with drift and a deterministic trend process, respectively.

Figure 8.3 Example of a white noise process

Figure 8.4 Time-series plot of a random walk versus a random walk with

444

drift

Figure 8.5 Time-series plot of a deterministic trend process

Comparing these three figures gives a good idea of the differences between the properties of a stationary, a stochastic trend and a deterministic trend process. In Figure 8.3, a white noise process visibly has no trending behaviour, and it frequently crosses its mean value of zero. The random walk (thick line) and random walk with drift (faint line) processes of Figure 8.4 exhibit ‘long swings’ away from their mean value, which they cross very rarely. A comparison of the two lines in this graph reveals that the positive drift leads to a series that is more likely to rise over time than to fall; obviously, the effect of the drift on the series becomes greater and greater the further the two processes are tracked. Finally, the deterministic trend process of Figure 8.5 clearly does not have a constant mean, and exhibits completely random fluctuations about its upward trend. If the trend were removed from the series, a plot similar to the white noise process of Figure 8.3 would result. In this author’s opinion, more time series in finance and economics look like Figure 8.4 than either Figure 8.3 or 8.5. Consequently, as stated above, the stochastic trend model will be the focus of the remainder of this chapter. Finally, Figure 8.6 plots the value of an autoregressive process of order 1 with different values of the autoregressive coefficient as given by equation (8.4). Values of ϕ = 0 (i.e., a white noise process), ϕ = 0.8 (i.e., a stationary AR(1)) and ϕ = 1 (i.e., a random walk) are plotted over time.

445

Figure 8.6 Autoregressive processes with differing values of ϕ (0, 0.8, 1)

8.1.3 Some More Definitions and Terminology If a non-stationary series, yt must be differenced d times before it becomes stationary, then it is said to be integrated of order d. This would be written yt ~ I(d). So if yt ~ I(d) then Δdyt ~ I(0). This latter piece of terminology states that applying the difference operator, Δ, d times, leads to an I(0) process, i.e., a process with no unit roots. In fact, applying the difference operator more than d times to an I(d) process will still result in a stationary series (but with an MA error structure). An I(0) series is a stationary series, while an I(1) series contains one unit root. For example, consider the random walk (8.24) An I(2) series contains two unit roots and so would require differencing twice to induce stationarity. I(1) and I(2) series can wander a long way from their mean value and cross this mean value rarely, while I(0) series should cross the mean frequently. The majority of financial and economic time series contain a single unit root, although some are stationary and some have been argued to possibly contain two unit roots (series such as nominal consumer prices and nominal wages). The efficient markets hypothesis together with rational expectations suggest that asset prices (or 446

the natural logarithms of asset prices) should follow a random walk or a random walk with drift, so that their differences are unpredictable (or only predictable to their long-term average value). To see what types of data generating process could lead to an I(2) series, consider the equation (8.25) taking all of the terms in y over to the LHS, and then applying the lag operator notation (8.26) (8.27) (8.28) It should be evident now that this process for yt contains two unit roots, and would require differencing twice to induce stationarity. What would happen if yt in equation (8.25) were differenced only once? Taking first differences of equation (8.25), i.e., subtracting yt−1 from both sides (8.29) (8.30) (8.31) (8.32) First differencing would therefore have removed one of the unit roots, but there is still a unit root remaining in the new variable, Δyt.

8.1.4 Testing for a Unit Root One immediately obvious (but inappropriate) method that readers may think of to test for a unit root would be to examine the autocorrelation function of the series of interest. However, although shocks to a unit root process will remain in the system indefinitely, the acf for a unit root process (a random walk) will often be seen to decay away very slowly to zero. Thus, such a process may be mistaken for a highly persistent but stationary process. Hence it is not possible to use the acf or pacf to determine whether a series is characterised by a unit root or not. 447

Furthermore, even if the true data generating process for yt contains a unit root, the results of the tests for a given sample could lead one to believe that the process is stationary. Therefore, what is required is some kind of formal hypothesis testing procedure that answers the question, ‘given the sample of data to hand, is it plausible that the true data generating process for y contains one or more unit roots?’ The early and pioneering work on testing for a unit root in time series was done by Dickey and Fuller (Fuller, 1976; Dickey and Fuller, 1979). The basic objective of the test is to examine the null hypothesis that ϕ = 1 in (8.33) against the one-sided alternative ϕ < 1. Thus the hypotheses of interest are H0: series contains a unit root versus H1: series is stationary. In practice, the following regression is employed, rather than equation (8.33), for ease of computation and interpretation (8.34) so that a test of ϕ = 1 is equivalent to a test of ψ = 0 (since ϕ − 1 = ψ). Dickey–Fuller (DF) tests are also known as τ-tests, and can be conducted allowing for an intercept, or an intercept and deterministic trend, or neither, in the test regression. The model for the unit root test in each case is (8.35) The tests can also be written, by subtracting yt−1 from each side of the equation, as (8.36) In another paper, Dickey and Fuller (1981) provide a set of additional test statistics and their critical values for joint tests of the significance of the lagged y, and the constant and trend terms. These are not examined further here. The test statistics for the original DF tests are defined as

448

(8.37)

The test statistics do not follow the usual t-distribution under the null hypothesis, since the null is one of non-stationarity, but rather they follow a non-standard distribution. Critical values are derived from simulations experiments in, for example, Fuller (1976); see also Chapter 13 in this book. Relevant examples of the distribution are shown in Table 8.1. A full set of DF critical values is given in the Appendix of Statistical Tables at the end of this book (Appendix 2). A discussion and example of how such critical values (CV) are derived using simulations methods are presented in Chapter 13. Table 8.1 Critical values for DF tests (Fuller, 1976, p. 373) Significance level

10%

5%

1%

CV for constant but no trend

−2.57

−2.86

−3.43

CV for constant and trend

−3.12

−3.41

−3.96

Comparing these with the standard normal critical values, it can be seen that the DF critical values are much bigger in absolute terms (i.e., more negative). Thus more evidence against the null hypothesis is required in the context of unit root tests than under standard t-tests. This arises partly from the inherent instability of the unit root process, the fatter distribution of the t-ratios in the context of non-stationary data (see Figure 8.2), and the resulting uncertainty in inference. The null hypothesis of a unit root is rejected in favour of the stationary alternative in each case if the test statistic is more negative than the critical value. The tests above are valid only if ut is white noise. In particular, ut is assumed not to be autocorrelated, but would be so if there was autocorrelation in the dependent variable of the regression (Δyt) which has not been modelled. If this is the case, the test would be ‘oversized’, meaning that the true size of the test (the proportion of times a correct null hypothesis is incorrectly rejected) would be higher than the nominal size used (e.g., 5%). The solution is to ‘augment’ the test using p lags of the dependent variable. The alternative model in case (i) (equation (8.34)) is 449

now written (8.38) The lags of Δyt now ‘soak up’ any dynamic structure present in the dependent variable, to ensure that ut is not autocorrelated. The test is known as an augmented Dickey–Fuller (ADF) test and is still conducted on ψ, and the same critical values from the DF tables are used as before. A problem now arises in determining the optimal number of lags of the dependent variable. Although several ways of choosing p have been proposed, they are all somewhat arbitrary, and are thus not presented here. Instead, the following two simple rules of thumb are suggested. First, the frequency of the data can be used to decide. So, for example, if the data are monthly, use twelve lags, if the data are quarterly, use four lags, and so on. Clearly, there would not be an obvious choice for the number of lags to use in a regression containing higher frequency financial data (e.g., hourly or daily)! Second, an information criterion can be used to decide. So choose the number of lags that minimises the value of an information criterion, as outlined in Chapter 7. It is quite important to attempt to use an optimal number of lags of the dependent variable in the test regression, and to examine the sensitivity of the outcome of the test to the lag length chosen. In most cases, hopefully the conclusion will not be qualitatively altered by small changes in p, but sometimes it will. Including too few lags will not remove all of the autocorrelation, thus biasing the results, while using too many will increase the coefficient standard errors. The latter effect arises since an increase in the number of parameters to estimate uses up degrees of freedom. Therefore, everything else being equal, the absolute values of the test statistics will be reduced. This will result in a reduction in the power of the test, implying that for a stationary process the null hypothesis of a unit root will be rejected less frequently than would otherwise have been the case.

8.1.5 Testing for Higher Orders of Integration Consider the simple regression (8.39)

450

H0: ψ = 0 is tested against H1: ψ < 0. If H0 is rejected, it would simply be concluded that yt does not contain a unit root. But what should be the conclusion if H0 is not rejected? The series contains a unit root, but is that it? No! What if yt ~ I(2)? The null hypothesis would still not have been rejected. It is now necessary to perform a test of Δ2yt(= Δyt − yt−1) would now be regressed on Δyt−1 (plus lags of Δ2yt to augment the test if necessary). Thus, testing H0: Δyt ~ I(1) is equivalent to H0: yt ~ I(2). So in this case, if H0 is not rejected (very unlikely in practice), it would be concluded that yt is at least I(2). If H0 is rejected, it would be concluded that yt contains a single unit root. The tests should continue for a further unit root until H0 is rejected. Dickey and Pantula (1987) have argued that an ordering of the tests as described above (i.e., testing for I(1), then I(2), and so on) is, strictly speaking, invalid. The theoretically correct approach would be to start by assuming some highest plausible order of integration (e.g., I(2)), and to test I(2) against I(1). If I(2) is rejected, then test I(1) against I(0). In practice, however, to the author’s knowledge, no financial time series contain more than a single unit root, so that this matter is of less concern in finance.

8.1.6 Phillips–Perron (PP) Tests Phillips and Perron have developed a more comprehensive theory of unit root non-stationarity. The tests are similar to ADF tests, but they incorporate an automatic correction to the DF procedure to allow for autocorrelated residuals. The tests often give the same conclusions as, and suffer from most of the same important limitations as, the ADF tests.

8.1.7 Criticisms of Dickey–Fuller- and Phillips–Perron-Type Tests The most important criticism that has been levelled at unit root tests is that their power is low if the process is stationary but with a root close to the non-stationary boundary. So, for example, consider an AR(1) data generating process with coefficient 0.95. If the true data generating process 451

is (8.40) the null hypothesis of a unit root should be rejected. It has been thus argued that the tests are poor at deciding, for example, whether ϕ = 1 or ϕ = 0.95, especially with small sample sizes. The source of this problem is that, under the classical hypothesis-testing framework, the null hypothesis is never accepted, it is simply stated that it is either rejected or not rejected. This means that a failure to reject the null hypothesis could occur either because the null was correct, or because there is insufficient information in the sample to enable rejection. One way to get around this problem is to use a stationarity test as well as a unit root test, as described in Box 8.1. BOX 8.1 Stationarity tests Stationarity tests have stationarity under the null hypothesis, thus reversing the null and alternatives under the Dickey–Fuller approach. Thus, under stationarity tests, the data will appear stationary by default if there is little information in the sample. One such stationarity test is the KPSS test (Kwaitkowski et al., 1992). The computation of the test statistic is not discussed here but the test is available within standard econometrics software such as EViews. The results of these tests can be compared with the ADF/PP procedure to see if the same conclusion is obtained. The null and alternative hypotheses under each testing approach are as follows:

There are four possible outcomes

For the conclusions to be robust, the results should fall under outcomes (1) or (2), which would be the case when both tests concluded that the series is stationary or non-stationary, respectively. Outcomes (3) or (4) imply conflicting results. The joint use of stationarity and unit root tests is known as confirmatory data analysis. 452

8.2 Tests for Unit Roots in the Presence of Structural Breaks 8.2.1 Motivation The standard Dickey-Fuller-type unit root tests presented above do not perform well if there are one or more structural breaks in the series under investigation, either in the intercept or the slope of the regression. More specifically, the tests have low power in such circumstances and they fail to reject the unit root null hypothesis when it is incorrect as the slope parameter in the regression of yt on yt−1 is biased towards unity by an unparameterised structural break. In general, the larger the break and the smaller the sample, the lower the power of the test. As Leybourne, Mills and Newbold (1998) have shown, unit root tests are also oversized in the presence of structural breaks, so they reject the null hypothesis too frequently when it is correct.1 Perron’s (1989) work is important since he was able to demonstrate that if we allow for structural breaks in the testing framework, a whole raft of macroeconomic series that Nelson and Plosser (1982) had identified as non-stationary may turn out to be stationary. He argues that most economic time series are best characterised by broken trend stationary processes, where the data generating process is a deterministic trend but with a structural break around 1929 that permanently changed the levels (i.e., the intercepts) of the series.

8.2.2 The Perron (1989) Procedure Recall from above that the flexible framework for unit root testing involves a regression of the form (8.41) where μ is an intercept and λt captures the time trend, one or both of which could be excluded from the regression if they were thought to be unnecessary. Perron (1989) proposes three test equations differing dependent on the type of break that was thought to be present. The first he terms a ‘crash’ model that allows a break in the level (i.e., the intercept) of the series; the 453

second is a ‘changing growth’ model that allows for a break in the growth rate (i.e., the slope) of the series; the final model allows for both types of break to occur at the same time, changing both the intercept and the slope of the trend. If we define the break point in the data as Tb, and Dt is a dummy variable defined as

the general equation for the third type of test (i.e., the most general) is (8.42) For the crash only model, set α2 = 0, while for the changing growth only model, set α1 = 0. In all three cases, there is a unit root with a structural break at Tb under the null hypothesis and a series that is a stationary process with a break under the alternative. While Perron (1989) commences a new literature on testing for unit roots in the presence of structural breaks, an important limitation of this approach is that it assumes that the break date is known in advance and the test is constructed using this information. It is possible, and perhaps even likely, however, that the date will not be known and must be determined from the data. More seriously, Christiano (1992) has argued that the critical values employed with the test will presume the break date to be chosen exogenously, and yet most researchers will select a break point based on an examination of the data and thus the asymptotic theory assumed will no longer hold. As a result, Banerjee, Lumsdaine and Stock (1992) and Zivot and Andrews (1992) introduce an approach to testing for unit roots in the presence of structural change that allows the break date to be selected endogenously. Their methods are based on recursive, rolling and sequential tests. For the recursive and rolling tests, Banerjee et al. propose four specifications. First, the standard Dickey–Fuller test on the whole sample, which they term second, the ADF test is conducted repeatedly on the sub-samples and the minimal DF statistic, is obtained; third, the maximal DF statistic is obtained from the sub-samples, finally, the difference between the maximal and minimal statistics, is taken. For the sequential test, the whole sample is used each time with the following regression being run 454

(8.43) where tused = Tb/T. The test is run repeatedly for different values of Tb over as much of the data as possible (a ‘trimmed sample’) that excludes the first few and the last few observations (since it is not possible to reliably detect breaks there). Clearly it is τt(tused) that allows for the break, which can either be in the level (where τt(tused) = 1 if t > tused and 0 otherwise); or the break can be in the deterministic trend (where τt(tused) = t − tused if t > tused and 0 otherwise). For each specification, a different set of critical values is required, and these can be found in Banerjee et al. (1992). Perron (1997) proposes an extension of the Perron (1989) technique but using a sequential procedure that estimates the test statistic allowing for a break at any point during the sample to be determined by the data. This technique is very similar to that of Zivot and Andrews, except that his is more flexible, and therefore arguably preferable, since it allows for a break under both the null and alternative hypotheses, whereas according to Zivot and Andrews’ model it can only arise under the alternative. A further extension would be to allow for more than one structural break in the series – for example, Lumsdaine and Papell (1997) enhance the Zivot and Andrews (1992) approach to allow for two structural breaks. It is also possible to allow for structural breaks in the cointegrating relationship between series (see Section 8.4 below for a thorough discussion of cointegration) using an extension of the first step in the Engle-Granger approach – see Gregory and Hansen (1996).

8.2.3 An Example: Testing for Unit Roots in EuroSterling Interest Rates Section 8.11 discusses the expectations hypothesis of the term structure of interest rates based on cointegration between the long and short rates. Clearly, key to this analysis is the question as to whether the interest rates themselves are I(1) or I(0) processes. Perhaps surprisingly, there is not a consensus in the empirical literature on whether this is the case. Brooks and Rew (2002) examine whether EuroSterling interest rates are best viewed as unit root process or not, allowing for the possibility of structural breaks in the series.2 They argue that failure to account for structural breaks that may be present in the data (caused, for example, by changes in monetary policy or the removal of exchange rate controls) may lead to 455

incorrect inferences regarding the validity or otherwise of the expectations hypothesis. Their sample covers the period 1 January 1981 to 1 September 1997 to total 4,348 data points. Brooks and Rew (2002) use the standard Dickey–Fuller test, the recursive and sequential tests of Banerjee et al. (1992), and their results are presented in Table 8.2. They also employ the rolling test, the Perron (1997) approach and several other techniques that are not shown here due to space limitations. Table 8.2 Recursive unit root tests for interest rates allowing for structural breaks

Notes: Source: Brooks and Garrett (2002), taken from Tables 1, 4 and 5. denotes the sequential test statistic allowing for a break in the trend, while is the test statistic allowing for a break in the level. The final row presents the 10% level critical values for each type of test obtained from Banerjee et al. (1992, p. 278, Table 2).

The findings for the recursive tests are the same as those for the standard DF test, and show that the unit root null should not be rejected at the 10% level for any of the maturities examined. For the sequential tests, the results are slightly more mixed with the break in trend model still showing no signs of rejecting the null hypothesis, while it is rejected for the short, seven-day and the one-month rates when a structural break is allowed for in the mean. Brooks and Rew’s overall conclusion is that the weight of evidence across all the tests they examine indicates that short term interest rates are best viewed as unit root processes that have a structural break in their level around the time of ‘Black Wednesday’ (16 September 1992) when the UK dropped out of the European Exchange Rate Mechanism (ERM). The 456

longer term-rates, on the other hand, are I(1) processes with no breaks.

8.2.4 Seasonal Unit Roots As we will discuss in detail in Chapter 10, many time series exhibit seasonal patterns. One approach to capturing such characteristics would be to use deterministic dummy variables at the frequency of the data (e.g., monthly dummy variables if the data are monthly). However, if the seasonal characteristics of the data are themselves changing over time so that their mean is not constant, then the use of dummy variables will be inadequate. Instead, we can entertain the possibility that a series may contain seasonal unit roots, so that it requires seasonal differencing to induce stationarity. We would use the notation I(d, D) to denote a series that is integrated of order d, D and requires differencing d times and seasonal differencing D times to obtain a stationary process. Osborn (1990) develops a test for seasonal unit roots based on a natural extension of the Dickey–Fuller approach. Groups of series with seasonal unit roots may also be seasonally cointegrated. However, Osborn also shows that only a small proportion of macroeconomic series exhibit seasonal unit roots; the majority have seasonal patterns that can better be characterised using dummy variables, which may explain why the concept of seasonal unit roots has not been widely adopted.3

8.3 Cointegration In most cases, if two variables that are I(1) are linearly combined, then the combination will also be I(1). More generally, if a set of variables Xi,t with differing orders of integration are combined, the combination will have an order of integration equal to the largest. If Xi,t ~ I(di) for i = 1, 2, 3, …, k so that there are k variables each integrated of order di, and letting (8.44) Then zt ~ I(max di). zt in this context is simply a linear combination of the k variables Xi. Rearranging equation (8.44) (8.45) 457

where All that has been done is to take one of the variables, X1, t, and to rearrange equation (8.44) to make it the subject. It could also be said that the equation has been normalised on X1,t. But viewed another way, equation (8.45) is just a regression equation where is a disturbance term. These disturbances would have some very undesirable properties: in general, will not be stationary and is autocorrelated if all of the Xi are I(1). As a further illustration, consider the following regression model containing variables yt, x2t, x3t which are all I(1) (8.46) For the estimated model, the SRF would be written (8.47) Taking everything except the residuals to the LHS (8.48) Again, the residuals when expressed in this way can be considered a linear combination of the variables. Typically, this linear combination of I(1) variables will itself be I(1), but it would obviously be desirable to obtain residuals that are I(0). Under what circumstances will this be the case? The answer is that a linear combination of I(1) variables will be I(0), in other words stationary, if the variables are cointegrated.

8.3.1 Definition of Cointegration (Engle and Granger, 1987) Let wt be a k × 1 vector of variables, then the components of wt are integrated of order (d, b) if (1) All components of wt are I(d) (2) There is at least one vector of coefficients α such that

In practice, many financial variables contain one unit root, and are thus I(1), so that the remainder of this chapter will restrict analysis to the case where d = b = 1. In this context, a set of variables is defined as 458

cointegrated if a linear combination of them is stationary. Many time series are non-stationary but ‘move together’ over time – that is, there exist some influences on the series (for example, market forces), which imply that the two series are bound by some relationship in the long run. A cointegrating relationship may also be seen as a long-term or equilibrium phenomenon, since it is possible that cointegrating variables may deviate from their relationship in the short run, but their association would return in the long run.

8.3.2 Examples of Possible Cointegrating Relationships in Finance Financial theory should suggest where two or more variables would be expected to hold some long-run relationship with one another. There are many examples in finance of areas where cointegration might be expected to hold, including Spot and futures prices for a given commodity or asset Ratio of relative prices and an exchange rate Equity prices and dividends. In all three cases, market forces arising from no-arbitrage conditions suggest that there should be an equilibrium relationship between the series concerned. The easiest way to understand this notion is perhaps to consider what would be the effect if the series were not cointegrated. If there were no cointegration, there would be no long-run relationship binding the series together, so that the series could wander apart without bound. Such an effect would arise since all linear combinations of the series would be non-stationary, and hence would not have a constant mean that would be returned to frequently. Spot and futures prices may be expected to be cointegrated since they are obviously prices for the same asset at different points in time, and hence will be affected in very similar ways by given pieces of information. The long-run relationship between spot and futures prices would be given by the cost of carry. Purchasing power parity (PPP) theory states that a given representative basket of goods and services should cost the same wherever it is bought when converted into a common currency. Further discussion of PPP occurs in Section 8.9, but for now suffice it to say that PPP implies that the ratio of relative prices in two countries and the exchange rate between them 459

should be cointegrated. If they did not cointegrate, assuming zero transactions costs, it would be profitable to buy goods in one country, sell them in another, and convert the money obtained back to the currency of the original country. Finally, if it is assumed that some stock in a particular company is held to perpetuity (i.e., for ever), then the only return that would accrue to that investor would be in the form of an infinite stream of future dividend payments. Hence the discounted dividend model argues that the appropriate price to pay for a share today is the present value of all future dividends. Hence, it may be argued that one would not expect current prices to ‘move out of line’ with future anticipated dividends in the long run, thus implying that share prices and dividends should be cointegrated. An interesting question to ask is whether a potentially cointegrating regression should be estimated using the levels of the variables or the logarithms of the levels of the variables. Financial theory may provide an answer as to the more appropriate functional form, but fortunately even if not, Hendry and Juselius (2000) note that if a set of series is cointegrated in levels, they will also be cointegrated in log levels.

8.4 Equilibrium Correction or Error Correction Models When the concept of non-stationarity was first considered in the 1970s, a usual response was to independently take the first differences of each of the I(1) variables and then to use these first differences in any subsequent modelling process. In the context of univariate modelling (e.g., the construction of ARMA models), this is entirely the correct approach. However, when the relationship between variables is important, such a procedure is inadvisable. While this approach is statistically valid, it does have the problem that pure first difference models have no long-run solution. For example, consider two series, yt and xt, that are both I(1). The model that one may consider estimating is (8.49) One definition of the long run that is employed in econometrics implies that the variables have converged upon some long-term values and are no longer changing, thus yt = yt−1 = y; xt = xt−1 = x. Hence all the difference terms will be zero in equation (8.49), i.e., Δyt = 0; Δxt = 0, and thus 460

everything in the equation cancels. Model equation (8.49) has no long-run solution and it therefore has nothing to say about whether x and y have an equilibrium relationship (see Chapter 5). Fortunately, there is a class of models that can overcome this problem by using combinations of first differenced and lagged levels of cointegrated variables. For example, consider the following equation (8.50) This model is known as an error correction model or an equilibrium correction model, and yt−1 − γ xt−1 is known as the error correction term. Provided that yt and xt are cointegrated with cointegrating coefficient γ, then (yt−1 − γ xt−1) will be I(0) even though the constituents are I(1). It is thus valid to use OLS and standard procedures for statistical inference on equation (8.50). It is of course possible to have an intercept in either the cointegrating term (e.g., yt−1 − α − γ xt−1) or in the model for Δyt (e.g., Δyt = β0 + β1 Δxt + β2(yt−1 − γ xt−1) + ut) or both. Whether a constant is included or not could be determined on the basis of financial theory, considering the arguments on the importance of a constant discussed in Chapter 5. The error correction model is sometimes termed an equilibrium correction model, and the two terms will be used synonymously for the purposes of this book. Error correction models are interpreted as follows. y is purported to change between t − 1 and t as a result of changes in the values of the explanatory variable(s), x, between t − 1 and t, and also in part to correct for any disequilibrium that existed during the previous period. Note that the error correction term (yt−1 − γ xt−1) appears in equation (8.50) with a lag. It would be implausible for the term to appear without any lag (i.e., as yt − γ xt), for this would imply that y changes between t − 1 and t in response to a disequilibrium at time t. γ defines the long-run relationship between x and y, while β1 describes the short-run relationship between changes in x and changes in y. Broadly, β2 describes the speed of adjustment back to equilibrium, and its strict definition is that it measures the proportion of last period’s equilibrium error that is corrected for. Of course, an error correction model can be estimated for more than two variables. For example, if there were three variables, xt, wt, yt, that were cointegrated, a possible error correction model would be 461

(8.51) The Granger representation theorem states that if there exists a dynamic linear model with stationary disturbances and the data are I(1), then the variables must be cointegrated of order (1,1).

8.5 Testing for Cointegration in Regression: A Residuals-Based Approach The model for the equilibrium correction term can be generalised further to include k variables (y and the k − 1 xs) (8.52) ut should be I(0) if the variables yt, x2t, …xkt are cointegrated, but ut will still be non-stationary if they are not. Thus it is necessary to test the residuals of equation (8.52) to see whether they are non-stationary or stationary. The DF or ADF test can be used on using a regression of the form (8.53) with vt an iid error term. However, since this is a test on residuals of a model, then the critical values are changed compared to a DF or an ADF test on a series of raw data. Engle and Granger (1987) have tabulated a new set of critical values for this application and hence the test is known as the Engle–Granger (EG) test. The reason that modified critical values are required is that the test is now operating on the residuals of an estimated model rather than on raw data. The residuals have been constructed from a particular set of coefficient estimates, and the sampling estimation error in those coefficients will change the distribution of the test statistic. Engle and Yoo (1987) tabulate a new set of critical values that are larger in absolute value (i.e., more negative) than the DF critical values, also given at the end of this book. The critical values also become more negative as the number of variables in the potentially cointegrating regression increases. It is also possible to use the Durbin–Watson (DW) test statistic or the Phillips–Perron (PP) approach to test for non-stationarity of If the DW test is applied to the residuals of the potentially cointegrating regression, it 462

is known as the Cointegrating Regression Durbin Watson (CRDW). Under the null hypothesis of a unit root in the errors, CRDW ≈ 0, so the null of a unit root is rejected if the CRDW statistic is larger than the relevant critical value (which is approximately 0.5). What are the null and alternative hypotheses for any unit root test applied to the residuals of a potentially cointegrating regression?

Thus, under the null hypothesis there is a unit root in the potentially cointegrating regression residuals, while under the alternative, the residuals are stationary. Under the null hypothesis, therefore, a stationary linear combination of the non-stationary variables has not been found. Hence, if this null hypothesis is not rejected, there is no cointegration. The appropriate strategy for econometric modelling in this case would be to employ specifications in first differences only. Such models would have no long-run equilibrium solution, but this would not matter since no cointegration implies that there is no long-run relationship anyway. On the other hand, if the null of a unit root in the potentially cointegrating regression’s residuals is rejected, it would be concluded that a stationary linear combination of the non-stationary variables had been found. Therefore, the variables would be classed as cointegrated. The appropriate strategy for econometric modelling in this case would be to form and estimate an error correction model, using a method described in the following section.

8.6 Methods of Parameter Estimation in Cointegrated Systems What should be the modelling strategy if the data at hand are thought to be non-stationary and possibly cointegrated? There are (at least) three methods that could be used: Engle–Granger, Engle–Yoo and Johansen. The first and third of these will be considered in some detail below.

8.6.1 The Engle–Granger 2-Step Method This is a single equation technique, which is conducted as follows: Step 1 463

Make sure that all the individual variables are I(1). Then estimate the cointegrating regression using OLS. Note that it is not possible to perform any inferences on the coefficient estimates in this regression – all that can be done is to estimate the parameter values. Save the residuals of the cointegrating regression, Test these residuals to ensure that they are I(0). If they are I(0), proceed to Step 2; if they are I(1), estimate a model containing only first differences. Step 2 Use the step 1 residuals as one variable in the error correction model, e.g., (8.54) where The stationary, linear combination of nonstationary variables is also known as the cointegrating vector. In this case, the cointegrating vector would be Additionally, any linear transformation of the cointegrating vector will also be a cointegrating vector. So, for example, will also be stationary. In equation (8.48) above, the cointegrating vector would be It is now valid to perform inferences in the second-stage regression, i.e., concerning the parameters β1 and β2 (provided that there are no other forms of misspecification, of course), since all variables in this regression are stationary. The Engle–Granger 2-step method suffers from a number of problems (1) The usual finite sample problem of a lack of power in unit root and cointegration tests discussed above. (2) There could be a simultaneous equations bias if the causality between y and x runs in both directions, but this single equation approach requires the researcher to normalise on one variable (i.e., to specify one variable as the dependent variable and the others as independent variables). The researcher is forced to treat y and x asymmetrically, even though there may have been no theoretical reason for doing so. A further issue is the following. Suppose that the following specification had been estimated as a potential cointegrating regression (8.55)

464

What if instead the following equation was estimated? (8.56) If it is found that u1t ~ I(0), does this imply automatically that u2t ~ I(0)? The answer in theory is ‘yes’, but in practice different conclusions may be reached in finite samples. Also, if there is an error in the model specification at stage 1, this will be carried through to the cointegration test at stage 2, as a consequence of the sequential nature of the computation of the cointegration test statistic. (3) It is not possible to perform any hypothesis tests about the actual cointegrating relationship estimated at stage 1. (4) There may be more than one cointegrating relationship – see Box 8.2. BOX 8.2 Multiple cointegrating relationships In the case where there are only two variables in an equation, yt, and xt, say, there can be at most only one linear combination of yt, and xt that is stationary – i.e., at most one cointegrating relationship. However, suppose that there are k variables in a system (ignoring any constant term), denoted yt, x2t, … xkt. In this case, there may be up to r linearly independent cointegrating relationships (where r ≤ k − 1). This potentially presents a problem for the OLS regression approach described above, which is capable of finding at most one cointegrating relationship no matter how many variables there are in the system. And if there are multiple cointegrating relationships, how can one know if there are others, or whether the ‘best’ or strongest cointegrating relationship has been found? An OLS regression will find the minimum variance stationary linear combination of the variables, but there may be other linear combinations of the variables that have more intuitive appeal. The answer to this problem is to use a systems approach to cointegration, which will allow determination of all r cointegrating relationships. One such approach is Johansen’s method – see Section 8.9. Problems (1) and (2) are small sample problems that should disappear asymptotically. Problem (3) is addressed by another method due to Engle and Yoo. There is also another alternative technique, which overcomes 465

problems (2) and (3) by adopting a different approach based on estimation of a VAR system – see Section 8.8.

8.6.2 The Engle and Yoo 3-Step Method The Engle and Yoo (1987) 3-step procedure takes its first two steps from Engle–Granger (EG). Engle and Yoo then add a third step giving updated estimates of the cointegrating vector and its standard errors. The Engle and Yoo (EY) third step is algebraically technical and additionally, EY suffers from all of the remaining problems of the EG approach. There is arguably a far superior procedure available to remedy the lack of testability of hypotheses concerning the cointegrating relationship – namely, the Johansen (1988) procedure. For these reasons, the Engle–Yoo procedure is rarely employed in empirical applications and is not considered further here. There now follows an application of the Engle–Granger procedure in the context of spot and futures markets.

8.7 Lead–Lag and Long-Term Relationships Between Spot and Futures Markets 8.7.1 Background If the markets are frictionless and functioning efficiently, changes in the (log of the) spot price of a financial asset and its corresponding changes in the (log of the) futures price would be expected to be perfectly contemporaneously correlated and not to be cross-autocorrelated. Mathematically, these notions would be represented as (a) (b) (c) In other words, changes in spot prices and changes in futures prices are expected to occur at the same time (condition (a)). The current change in the futures price is also expected not to be related to previous changes in the spot price (condition (b)), and the current change in the spot price is expected not to be related to previous changes in the futures price 466

(condition (c)). The changes in the log of the spot and futures prices are also of course known as the spot and futures returns. For the case when the underlying asset is a stock index, the equilibrium relationship between the spot and futures prices is known as the cost of carry model, given by (8.57) where is the fair futures price, St is the spot price, r is a continuously compounded risk-free rate of interest, d is the continuously compounded yield in terms of dividends derived from the stock index until the futures contract matures, and (T − t) is the time to maturity of the futures contract. Taking logarithms of both sides of (8.57) gives (8.58) where is the log of the fair futures price and st is the log of the spot price. Equation (8.58) suggests that the long-term relationship between the logs of the spot and futures prices should be one to one. Thus the basis, defined as the difference between the futures and spot prices (and if necessary adjusted for the cost of carry) should be stationary, for if it could wander without bound, arbitrage opportunities would arise, which would be assumed to be quickly acted upon by traders such that the relationship between spot and futures prices will be brought back to equilibrium. The notion that there should not be any lead–lag relationships between the spot and futures prices and that there should be a long-term one to one relationship between the logs of spot and futures prices can be tested using simple linear regressions and cointegration analysis. This book will now examine the results of two related papers – Tse (1995), who employs daily data on the Nikkei Stock Average (NSA) and its futures contract, and Brooks, Brooks, Rew, and Ritson (2001), who examine high-frequency data from the FTSE 100 stock index and index futures contract. The data employed by Tse (1995) consists of 1,055 daily observations on NSA stock index and stock index futures values from December 1988 to April 1993. The data employed by Brooks et al. comprises 13,035 tenminutely observations for all trading days in the period June 1996–May 1997, provided by FTSE International. In order to form a statistically adequate model, the variables should first be checked as to whether they can be considered stationary. The results of applying a DF test to the logs 467

of the spot and futures prices of the ten-minutely FTSE data are shown in Table 8.3. Table 8.3 DF tests on log-prices and returns for high frequency FTSE data Futures Dickey–Fuller statistics for logprice data Dickey–Fuller statistics for returns data

Spot

−0.1329

−0.7335

−84.9968

−114.1803

As one might anticipate, both studies conclude that the two log–price series contain a unit root, while the returns are stationary. Of course, it may be necessary to augment the tests by adding lags of the dependent variable to allow for autocorrelation in the errors (i.e., an ADF test). Results for such tests are not presented, since the conclusions are not altered. A statistically valid model would therefore be one in the returns. However, a formulation containing only first differences has no long-run equilibrium solution. Additionally, theory suggests that the two series should have a long-run relationship. The solution is therefore to see whether there exists a cointegrating relationship between ft and st which would mean that it is valid to include levels terms along with returns in this framework. This is tested by examining whether the residuals, of a regression of the form (8.59) are stationary, using a DF test, where zt is the error term. The coefficient values for the estimated equation (8.59) and the DF test statistic are given in Table 8.4. Table 8.4 Estimated potentially cointegrating equation and test for cointegration for high frequency FTSE data Coefficient

Estimated value 0.1345 468

0.9834 DF test on residuals

Test statistic −14.7303

Source: Brooks, Rew, and Ritson (2001).

Clearly, the residuals from the cointegrating regression can be considered stationary. Note also that the estimated slope coefficient in the cointegrating regression takes on a value close to unity, as predicted from the theory. It is not possible to formally test whether the true population coefficient could be one, however, since there is no way in this framework to test hypotheses about the cointegrating relationship. The final stage in building an error correction model using the Engle– Granger two-step approach is to use a lag of the first-stage residuals, as the equilibrium correction term in the general equation. The overall model is (8.60) where vt is an error term. The coefficient estimates for this model are presented in Table 8.5. Table 8.5 Estimated error correction model for high frequency FTSE data Coefficient

Estimated value

t-ratio

9.6713E−06

1.6083

−0.8388

−5.1298

0.1799

19.2886

0.1312

20.4946

Source: Brooks, Rew, and Ritson (2001).

Consider first the signs and significances of the coefficients (these can now be interpreted validly since all variables used in this model are stationary). is positive and highly significant, indicating that the futures market does indeed lead the spot market, since lagged changes in futures prices lead to a positive change in the subsequent spot price. is positive and highly significant, indicating on average a positive autocorrelation in spot returns. the coefficient on the error correction term, is negative and 469

significant, indicating that if the difference between the logs of the spot and futures prices is positive in one period, the spot price will fall during the next period to restore equilibrium, and vice versa.

8.7.2 Forecasting Spot Returns Both Brooks, Rew, and Ritson (2001) and Tse (1995) show that it is possible to use an error correction formulation to model changes in the log of a stock index. An obvious related question to ask is whether such a model can be used to forecast the future value of the spot series for a holdout sample of data not used previously for model estimation. Both sets of researchers employ forecasts from three other models for comparison with the forecasts of the error correction model. These are an error correction model with an additional term that allows for the cost of carry, an ARMA model (with lag length chosen using an information criterion) and an unrestricted VAR model (with lag length chosen using a multivariate information criterion). The results are evaluated by comparing their root-mean squared errors, mean absolute errors and percentage of correct direction predictions. The forecasting results from the Brooks, Rew and Ritson paper are given in Table 8.6. Table 8.6 Comparison of out-of-sample forecasting accuracy ECM

ECM-COC

ARIMA

VAR

RMSE

0.0004382

0.0004350

0.0004531

0.0004510

MAE

0.4259

0.4255

0.4382

0.4378

% Correct direction

67.69%

68.75%

64.36%

66.80%

Source: Brooks, Rew, and Ritson (2001).

It can be seen from Table 8.6 that the error correction models have both the lowest mean squared and mean absolute errors, and the highest proportion of correct direction predictions. There is, however, little to choose between the models, and all four have over 60% of the signs of the next returns predicted correctly. It is clear that on statistical grounds the out-of-sample forecasting performances of the error correction models are better than those of their competitors, but this does not necessarily mean that such forecasts have 470

any practical use. Many studies have questioned the usefulness of statistical measures of forecast accuracy as indicators of the profitability of using these forecasts in a practical trading setting (see, for example, Leitch and Tanner, 1991). Brooks, Rew, and Ritson (2001) investigate this proposition directly by developing a set of trading rules based on the forecasts of the error correction model with the cost of carry term, the best statistical forecasting model. The trading period is an out-of-sample data series not used in model estimation, running from 1 May–30 May 1997. The error correction model with cost of carry (ECM-COC) model yields ten-minutely one-step-ahead forecasts. The trading strategy involves analysing the forecast for the spot return, and incorporating the decision dictated by the trading rules described below. It is assumed that the original investment is £1,000, and if the holding in the stock index is zero, the investment earns the risk-free rate. Five trading strategies are employed, and their profitabilities are compared with that obtained by passively buying and holding the index. There are of course an infinite number of strategies that could be adopted for a given set of spot return forecasts, but Brooks, Rew and Ritson use the following Liquid trading strategy This trading strategy involves making a round-trip trade (i.e., a purchase and sale of the FTSE 100 stocks) every ten minutes that the return is predicted to be positive by the model. If the return is predicted to be negative by the model, no trade is executed and the investment earns the risk-free rate. Buy-and-hold while forecast positive strategy This strategy allows the trader to continue holding the index if the return at the next predicted investment period is positive, rather than making a roundtrip transaction for each period. Filter strategy: better predicted return than average This strategy involves purchasing the index only if the predicted returns are greater than the average positive return (there is no trade for negative returns therefore the average is only taken of the positive returns). Filter strategy: better predicted return than first decile This strategy is similar to the previous one, but rather than utilising the average as previously, only the returns predicted to be in the top 10% of all returns are traded on. Filter strategy: high arbitrary cutoff An arbitrary filter of 0.0075% is imposed, which will result in trades only for returns that are predicted to be extremely large for a ten-minute interval.

471

The results from employing each of the strategies using the forecasts for the spot returns obtained from the ECM-COC model are presented in Table 8.7. Table 8.7 Trading profitability of the error correction model with cost of carry

Source: Brooks, Rew, and Ritson (2001).

The test month of May 1997 was a particularly bullish one, with a pure buy-and-hold-the-index strategy netting a return of 4%, or almost 50% on an annualised basis. Ideally, the forecasting exercise would be conducted over a much longer period than one month, and preferably over different market conditions. However, this was simply impossible due to the lack of availability of very high frequency data over a long time period. Clearly, the forecasts have some market timing ability in the sense that they seem to ensure trades that, on average, would have invested in the index when it rose, but be out of the market when it fell. The most profitable trading strategies in gross terms are those that trade on the basis of every positive spot return forecast, and all rules except the strictest filter make more money than a passive investment. The strict filter appears not to work well since it is out of the index for too long during a period when the market is rising strongly. However, the picture of immense profitability painted thus far is 472

somewhat misleading for two reasons: slippage time and transactions costs. First, it is unreasonable to assume that trades can be executed in the market the minute they are requested, since it may take some time to find counterparties for all the trades required to ‘buy the index’. (Note, of course, that in practice, a similar returns profile to the index can be achieved with a very much smaller number of stocks.) Brooks, Rew and Ritson therefore allow for ten minutes of ‘slippage time’, which assumes that it takes ten minutes from when the trade order is placed to when it is executed. Second, it is unrealistic to consider gross profitability, since transactions costs in the spot market are non-negligible and the strategies examined suggested a lot of trades. Sutcliffe (1997, p. 47) suggests that total round-trip transactions costs for FTSE stocks are of the order of 1.7% of the investment. The effect of slippage time is to make the forecasts less useful than they would otherwise have been. For example, if the spot price is forecast to rise, and it does, it may have already risen and then stopped rising by the time that the order is executed, so that the forecasts lose their market timing ability. Terminal wealth appears to fall substantially when slippage time is allowed for, with the monthly return falling by between 1.5% and 10%, depending on the trading rule. Finally, if transactions costs are allowed for, none of the trading rules can outperform the passive investment strategy, and all in fact make substantial losses.

8.7.3 Conclusions If the markets are frictionless and functioning efficiently, changes in the spot price of a financial asset and its corresponding futures price would be expected to be perfectly contemporaneously correlated and not to be crossautocorrelated. Many academic studies, however, have documented that the futures market systematically ‘leads’ the spot market, reflecting news more quickly as a result of the fact that the stock index is not a single entity. The latter implies that Some components of the index are infrequently traded, implying that the observed index value contains ‘stale’ component prices It is more expensive to transact in the spot market and hence the spot market reacts more slowly to news Stock market indices are recalculated only every minute so that new information takes longer to be reflected in the index. 473

Clearly, such spot market impediments cannot explain the inter-daily lead–lag relationships documented by Tse (1995). In any case, however, since it appears impossible to profit from these relationships, their existence is entirely consistent with the absence of arbitrage opportunities and is in accordance with modern definitions of the efficient markets hypothesis.

8.8 Testing for and Estimating Cointegration in Systems Using the Johansen Technique based on VARs Suppose that a set of g variables (g ≥ 2) are under consideration that are I(1) and which are thought may be cointegrated. A VAR with k lags containing these variables could be set up: (8.61) In order to use the Johansen test, the VAR (8.61) above needs to be turned into a vector error correction model (VECM) of the form (8.62) where and This VAR contains g variables in first differenced form on the LHS, and k − 1 lags of the dependent variables (differences) on the RHS, each with a Γ coefficient matrix attached to it. In fact, the Johansen test can be affected by the lag length employed in the VECM, and so it is useful to attempt to select the lag length optimally, as outlined in Chapter 7. The Johansen test centres around an examination of the Π matrix. Π can be interpreted as a long-run coefficient matrix, since in equilibrium, all the Δyt−i will be zero, and setting the error terms, ut, to their expected value of zero will leave Πyt−k = 0. Notice the comparability between this set of equations and the testing equation for an ADF test, which has a first differenced term as the dependent variable, together with a lagged levels term and lagged differences on the RHS. The test for cointegration between the ys is calculated by looking at the rank of the Π matrix via its eigenvalues.2 The rank of a matrix is equal to the number of its characteristic roots (eigenvalues) that are different from 474

zero (see Section 1.7.5 for some algebra and examples). The eigenvalues, denoted λi are put in descending order λ1 ≥ λ2 ≥ … ≥ λg. If the λs are roots, in this context they must be less than one in absolute value and positive, and λ1 will be the largest (i.e., the closest to one), while λg will be the smallest (i.e., the closest to zero). If the variables are not cointegrated, the rank of will not be significantly different from zero, so λi ≈ 0 ∀ i. The test statistics actually incorporate ln(1 − λi), rather than the λi themselves, but still, when λi = 0, ln(1 − λi) = 0. Suppose now that rank (Π) = 1, then ln(1 − λ1) will be negative and ln(1 − λi) = 0∀i > 1. If the eigenvalue i is non-zero, then ln(1 − λi) < 0∀i ≥ 1. That is, for to have a rank of 1, the largest eigenvalue must be significantly non-zero, while others will not be significantly different from zero. There are two test statistics for cointegration under the Johansen approach, which are formulated as (8.63) and (8.64) where r is the number of cointegrating vectors under the null hypothesis and is the estimated value for the ith ordered eigenvalue from the Π matrix. Intuitively, the larger is the more large and negative will be and hence the larger will be the test statistic. Each eigenvalue will have associated with it a different cointegrating vector, which will be an eigenvector. A significantly non-zero eigenvalue indicates a significant cointegrating vector. λtrace is a joint test where the null is that the number of cointegrating vectors is less than or equal to r against an unspecified or general alternative that there are more than r. It starts with p eigenvalues, and then successively the largest is removed. λtrace = 0 when all the λi = 0, for i = 1, …, g. λmax conducts separate tests on each eigenvalue, and has as its null hypothesis that the number of cointegrating vectors is r against an alternative of r + 1. Johansen and Juselius (1990) provide critical values for the two 475

statistics. The distribution of the test statistics is non-standard, and the critical values depend on the value of g − r, the number of non-stationary components and whether constants are included in each of the equations. Intercepts can be included either in the cointegrating vectors themselves or as additional terms in the VAR. The latter is equivalent to including a trend in the data generating processes for the levels of the series. Osterwald-Lenum (1992) provides a more complete set of critical values for the Johansen test, some of which are also given in the Appendix of Statistical Tables (Appendix 2) at the end of this book. If the test statistic is greater than the critical value from Johansen’s tables, reject the null hypothesis that there are r cointegrating vectors in favour of the alternative that there are r + 1 (for λmax) or more than r (for λtrace). The testing is conducted in a sequence and under the null, r = 0, 1, …, g − 1 so that the hypotheses for λtrace are

The first test involves a null hypothesis of no cointegrating vectors (corresponding to Π having zero rank). If this null is not rejected, it would be concluded that there are no cointegrating vectors and the testing would be completed. However, if H0 : r = 0 is rejected, the null that there is one cointegrating vector (i.e., H0 : r = 1) would be tested and so on. Thus the value of r is continually increased until the null is no longer rejected. But how does this correspond to a test of the rank of the Π matrix? r is the rank of Π. Π cannot be of full rank (g) since this would correspond to the original yt being stationary. If Π has zero rank, then by analogy to the univariate case, Δyt depends only on Δyt−j and not on yt−1, so that there is no long-run relationship between the elements of yt−1. Hence there is no cointegration. For 1 ≤ rank(Π) < g, there are r cointegrating vectors. Π is then defined as the product of two matrices, α and β′, of dimension (g × r) and (r × g), respectively, i.e., (8.65) The matrix β gives the cointegrating vectors, while α gives the amount of 476

each cointegrating vector entering each equation of the VECM, also known as the ‘adjustment parameters’. For example, suppose that g = 4, so that the system contains four variables. The elements of the Π matrix would be written

(8.66)

If r = 1, so that there is one cointegrating vector, then α and β will be (4 × 1)

(8.67)

If r = 2, so that there are two cointegrating vectors, then α and β will be (4 × 2)

(8.68)

and so on for r = 3, … Suppose now that g = 4, and r = 1, as in equation (8.67), so that there are four variables in the system, y1, y2, y3, and y4, that exhibit one cointegrating vector. Then Πyt−k will be given by

(8.69)

Equation (8.69) can also be written

477

(8.70)

Given equation (8.70), it is possible to write out the separate equations for each variable Δyt. It is also common to ‘normalise’ on a particular variable, so that the coefficient on that variable in the cointegrating vector is one. For example, normalising on y1 would make the cointegrating term in the equation for Δy1

Finally, it must be noted that the above description is not exactly how the Johansen procedure works, but is an intuitive approximation to it.

8.8.1 Tests for Cointegration with Mixed Orders of Integration Suppose that we have a set of variables which we believe are related to one another and where there may potentially be a long-term relationship between some of them but where the individual variables are of different orders of integration. In the context of the Engle-Granger single equation approach, the test for cointegration will still be applicable, but the order of integration of the residuals in the potentially cointegrating regression will be the highest of the individual variables if they are not cointegrated and I(0) if they are cointegrated. In practice we will again only be considering variables that are either I(1) or I(0), so suppose we have a set of three variables which are individually I(1), I(1), and I(0). If the variables are cointegrated then the residuals will be I(0) since these residuals will be a stationary linear combination of the two I(1) variables and the variable which was already stationary (I(0)), whereas if they are not cointegrated then the residuals will be I(1). Thus the I(0) variable effectively acts like a constant from the perspective of non-stationarity. Within the Johansen framework, if the number of variables in the system is N, then the cointegrating rank is equal to the sum of the number of linearly independent cointegrating vectors and the number of I(0) variables in the system.

8.8.2 Hypothesis Testing using Johansen 478

Engle–Granger did not permit the testing of hypotheses on the cointegrating relationships themselves, but the Johansen setup does permit the testing of hypotheses about the equilibrium relationships between the variables. Johansen allows a researcher to test a hypothesis about one or more coefficients in the cointegrating relationship by viewing the hypothesis as a restriction on the Π matrix. If there exist r cointegrating vectors, only these linear combinations or linear transformations of them, or combinations of the cointegrating vectors, will be stationary. In fact, the matrix of cointegrating vectors β can be multiplied by any non-singular conformable matrix to obtain a new set of cointegrating vectors. A set of required long-run coefficient values or relationships between the coefficients does not necessarily imply that the cointegrating vectors have to be restricted. This is because any combination of cointegrating vectors is also a cointegrating vector. So it may be possible to combine the cointegrating vectors thus far obtained to provide a new one or, in general, a new set, having the required properties. The simpler and fewer are the required properties, the more likely that this recombination process (called renormalisation) will automatically yield cointegrating vectors with the required properties. However, as the restrictions become more numerous or involve more of the coefficients of the vectors, it will eventually become impossible to satisfy all of them by renormalisation. After this point, all other linear combinations of the variables will be non-stationary. If the restriction does not affect the model much, i.e., if the restriction is not binding, then the eigenvectors should not change much following imposition of the restriction. A test statistic to test this hypothesis is given by (8.71) where are the characteristic roots of the restricted model, λi are the characteristic roots of the unrestricted model, r is the number of non-zero characteristic roots in the unrestricted model and m is the number of overidentifying restrictions. Restrictions are actually imposed by substituting them into the relevant α or β matrices as appropriate, so that tests can be conducted on either the cointegrating vectors or their loadings in each equation in the system (or both). For example, considering equations (8.66)–(8.68) above, it may be that theory suggests that the coefficients on the loadings of the cointegrating vector(s) in each equation should take on certain values, in 479

which case it would be relevant to test restrictions on the elements of α (e.g. α11 = 1, α23 = −1, etc.). Equally, it may be of interest to examine whether only a sub-set of the variables in yt is actually required to obtain a stationary linear combination. In that case, it would be appropriate to test restrictions of elements of β. For example, to test the hypothesis that y4 is not necessary to form a long-run relationship, set β14 = 0, β24 = 0, etc. For an excellent detailed treatment of cointegration in the context of both single equation and multiple equation models, see Harris (1995). Several applications of tests for cointegration and modelling cointegrated systems in finance will now be given.

8.9 Purchasing Power Parity Purchasing power parity (PPP) states that the equilibrium or long-run exchange rate between two countries is equal to the ratio of their relative price levels. Purchasing power parity implies that the real exchange rate, Qt, is stationary. The real exchange rate can be defined as (8.72) where Et is the nominal exchange rate in domestic currency per unit of foreign currency, Pt is the domestic price level and Pt* is the foreign price level. Taking logarithms of equation (8.72) and rearranging, another way of stating the PPP relation is obtained (8.73) where the lower case letters in equation (8.73) denote logarithmic transforms of the corresponding upper case letters used in equation (8.72). A necessary and sufficient condition for PPP to hold is that the variables on the LHS of equation (8.73) – that is the log of the exchange rate between countries A and B, and the logs of the price levels in countries A and B be cointegrated with cointegrating vector [1 – 1 1]. A test of this form is conducted by Chen (1995) using monthly data from Belgium, France, Germany, Italy and the Netherlands over the period April 1973 to December 1990. Pair-wise evaluations of the existence or otherwise of cointegration are examined for all combinations of these countries (ten country pairs). Since there are three variables in the system 480

(the log exchange rate and the two log nominal price series) in each case, and that the variables in their log-levels forms are nonstationary, there can be at most two linearly independent cointegrating relationships for each country pair. The results of applying Johansen’s trace test are presented in Chen’s Table 1, adapted and presented here as Table 8.8. Table 8.8 Cointegration tests of PPP with European data Tests for cointegration between

r=0

r≤1

r≤2

α1

α2

FRF–DEM

34.63*

17.10

6.26

1.33

−2.50

FRF–ITL

52.69*

15.81

5.43

2.65

−2.52

FRF–NLG

68.10*

16.37

6.42

0.58

−0.80

FRF–BEF

52.54*

26.09*

3.63

0.78

−1.15

DEM–ITL

42.59*

20.76*

4.79

5.80

−2.25

DEM–NLG

50.25*

17.79

3.28

0.12

−0.25

DEM–BEF

69.13*

27.13*

4.52

0.87

−0.52

ITL–NLG

37.51*

14.22

5.05

0.55

−0.71

ITL–BEF

69.24*

32.16*

7.15

0.73

−1.28

NLG–BEF

64.52*

21.97*

3.88

1.69

−2.17

Critical values

31.52

17.95

8.18

–

–

Notes: FRF – French franc; DEM – German mark; NLG – Dutch guilder; ITL – Italian lira; BEF – Belgian franc. Source: Chen (1995). Reprinted with the permission of Taylor and Francis Ltd (www.tandf.co.uk).

As can be seen from the results, the null hypothesis of no cointegrating vectors is rejected for all country pairs, and the null of one or fewer cointegrating vectors is rejected for France–Belgium, Germany–Italy, Germany–Belgium, Italy–Belgium, Netherlands–Belgium. In no cases is the null of two or less cointegrating vectors rejected. It is therefore concluded that the PPP hypothesis is upheld and that there are either one or two cointegrating relationships between the series depending on the 481

country pair. Estimates of α1 and α2 are given in the last two columns of Table 8.8. PPP suggests that the estimated values of these coefficients should be 1 and −1, respectively. In most cases, the coefficient estimates are a long way from these expected values. Of course, it would be possible to impose this restriction and to test it in the Johansen framework as discussed above, but Chen does not conduct this analysis.

8.10 Cointegration Between International Bond Markets Often, investors will hold bonds from more than one national market in the expectation of achieving a reduction in risk via the resulting diversification. If international bond markets are very strongly correlated in the long run, diversification will be less effective than if the bond markets operated independently of one another. An important indication of the degree to which long-run diversification is available to international bond market investors is given by determining whether the markets are cointegrated. This book will now study two examples from the academic literature that consider this issue: Clare, Maras and Thomas (1995), and Mills and Mills (1991).

8.10.1 Cointegration Between International Bond Markets: A Univariate Approach Clare, Maras and Thomas (1995) use the Dickey–Fuller and Engle– Granger single-equation method to test for cointegration using a pair-wise analysis of four countries’ bond market indices: US, UK, Germany and Japan. Monthly Salomon Brothers’ total return government bond index data from January 1978 to April 1990 are employed. An application of the Dickey–Fuller test to the log of the indices reveals the following results (adapted from their Table 1), given in Table 8.9. Table 8.9 DF tests for international bond indices Panel A: test on log-index for country

DF Statistic

Germany

−0.395

Japan

−0.799 482

UK

−0.884

US

0.174

Panel B: test on log-returns for country Germany

−10.37

Japan

−10.11

UK

−10.56

US

−10.64

Source: Clare, Maras and Thomas (1995). Reprinted with the permission of Blackwell Publishers.

Neither the critical values, nor a statement of whether a constant or trend are included in the test regressions, are offered in the paper. Nevertheless, the results are clear. Recall that the null hypothesis of a unit root is rejected if the test statistic is smaller (more negative) than the critical value. For samples of the size given here, the 5% critical value would be somewhere between −1.95 and −3.50. It is thus demonstrated quite conclusively that the logarithms of the indices are non-stationary, while taking the first difference of the logs (that is, constructing the returns) induces stationarity. Given that all logs of the indices in all four cases are shown to be I(1), the next stage in the analysis is to test for cointegration by forming a potentially cointegrating regression and testing its residuals for nonstationarity. Clare, Maras and Thomas use regressions of the form (8.74) with time subscripts suppressed and where Bi and Bj represent the logbond indices for any two countries i and j. The results are presented in their Tables 3 and 4, which are combined into Table 8.10 here. They offer findings from applying seven different tests, while we present the results for only the Cointegrating Regression Durbin Watson (CRDW), Dickey– Fuller and Augmented Dickey–Fuller tests (although the lag lengths for the latter are not given in their paper). Table 8.10 Cointegration tests for pairs of international bond indices

483

Test

UK– Germany

UK– Japan

UK– US

CRDW

0.189

0.197

DF

2.970

ADF

3.160

Germany– Japan

Germany– US

Japan– US

0.097 0.230

0.169

0.139

2.770

2.020 3.180

2.160

2.160

2.900

1.800 3.360

1.640

1.890

Source: Clare, Maras and Thomas (1995). Reprinted with the permission of Blackwell Publishers.

In this case, the null hypothesis of a unit root in the residuals from regression (8.74) cannot be rejected. The conclusion is therefore that there is no cointegration between any pair of bond indices in this sample.

8.10.2 Cointegration Between International Bond Markets: A Multivariate Approach Mills and Mills (1991) also consider the issue of cointegration or noncointegration between the same four international bond markets. However, unlike Clare et al. (1995), who use bond price indices, Mills and Mills employ daily closing observations on the redemption yields. The latter’s sample period runs from 1 April 1986 to 29 December 1989, giving 960 observations. They employ a Dickey–Fuller-type regression procedure to test the individual series for non-stationarity and conclude that all four yields series are I(1). The Johansen systems procedure is then used to test for cointegration between the series. Unlike Clare et al., Mills and Mills consider all four indices together rather than investigating them in a pair-wise fashion. Therefore, since there are four variables in the system (the redemption yield for each country), i.e., g = 4, there can be at most three linearly independent cointegrating vectors, i.e., r ≤ 3. The trace statistic is employed, and it takes the form (8.75) where λi are the ordered eigenvalues. The results are presented in their Table 2, which is modified slightly here, and presented in Table 8.11. Table 8.11 Johansen tests for cointegration between international bond yields 484

r (number of cointegrating

Critical values

vectors under the null hypothesis)

Test statistic

10%

5%

0

22.06

35.6

38.6

1

10.58

21.2

23.8

2

2.52

10.3

12.0

3

0.12

2.9

4.2

Source: Mills and Mills (1991). Reprinted with the permission of Blackwell Publishers.

Looking at the first row under the heading, it can be seen that the test statistic is smaller than the critical value, so the null hypothesis that r = 0 cannot be rejected, even at the 10% level. It is thus not necessary to look at the remaining rows of the table. Hence, reassuringly, the conclusion from this analysis is the same as that of Clare et al. – i.e., that there are no cointegrating vectors. Given that there are no linear combinations of the yields that are stationary, and therefore that there is no error correction representation, Mills and Mills then continue to estimate a VAR for the first differences of the yields. The VAR is of the form (8.76) where

They set k, the number of lags of each change in the yield in each regression, to 8, arguing that likelihood ratio tests rejected the possibility of smaller numbers of lags. Unfortunately, and as one may anticipate for a regression of daily yield changes, the R2 values for the VAR equations are low, ranging from 0.04 for the US to 0.17 for Germany. Variance decompositions and impulse responses are calculated for the estimated VAR. Two orderings of the variables are employed: one based on a 485

previous study and one based on the chronology of the opening (and closing) of the financial markets considered: Japan → Germany → UK → US. Only results for the latter, adapted from Tables 4 and 5 of Mills and Mills (1991), are presented here. The variance decompositions and impulse responses for the VARs are given in Tables 8.12 and 8.13, respectively. Table 8.12 Variance decompositions for VAR of international bond yields

Source: Mills and Mills (1991). Reprinted with the permission of Blackwell Publishers.

Table 8.13 Impulse responses for VAR of international bond yields

486

487

Source: Mills and Mills (1991). Reprinted with the permission of Blackwell Publishers.

As one may expect from the low R2 of the VAR equations, and the lack of cointegration, the bond markets seem very independent of one another. The variance decompositions, which show the proportion of the movements in the dependent variables that are due to their ‘own’ shocks, versus shocks to the other variables, seem to suggest that the US, UK and Japanese markets are to a certain extent exogenous in this system. That is, little of the movement of the US, UK or Japanese series can be explained by movements other than their own bond yields. In the German case, however, after twenty days, only 83% of movements in the German yield are explained by German shocks. The German yield seems particularly influenced by US (8.4% after twenty days) and UK (6.5% after twenty days) shocks. It also seems that Japanese shocks have the least influence on the bond yields of other markets. A similar pattern emerges from the impulse response functions, which show the effect of a unit shock applied separately to the error of each equation of the VAR. The markets appear relatively independent of one another, and also informationally efficient in the sense that shocks work through the system very quickly. There is never a response of more than 10% to shocks in any series three days after they have happened; in most cases, the shocks have worked through the system in two days. Such a result implies that the possibility of making excess returns by trading in one market on the basis of ‘old news’ from another appears very unlikely.

8.10.3 Cointegration in International Bond Markets: Conclusions A single set of conclusions can be drawn from both of these papers. Both approaches have suggested that international bond markets are not cointegrated. This implies that investors can gain substantial diversification benefits. This is in contrast to results reported for other markets, such as foreign exchange (Baillie and Bollerslev, 1989), commodities (Baillie, 1989) and equities (Taylor and Tonks, 1989). Clare, Maras and Thomas (1995) suggest that the lack of long-term integration between the markets may be due to ‘institutional idiosyncrasies’, such as heterogeneous maturity and taxation structures, and differing investment cultures, issuance patterns and macroeconomic policies between countries, which imply that the markets operate largely independently of one another. 488

8.11 Testing the Expectations Hypothesis of the Term Structure of Interest Rates The following notation replicates that employed by Campbell and Shiller (1991) in their seminal paper. The single, linear expectations theory of the term structure used to represent the expectations hypothesis (hereafter EH), defines a relationship between an n-period interest rate or yield, denoted and an m-period interest rate, denoted where n > m. Hence is the interest rate or yield on a longer-term instrument relative to a shorter-term interest rate or yield, More precisely, the EH states that the expected return from investing in an n-period rate will equal the expected return from investing in m-period rates up to n − m periods in the future plus a constant risk-premium, c, which can be expressed as (8.77) where q = n/m. Consequently, the longer-term interest rate, can be expressed as a weighted-average of current and expected shorter-term interest rates, plus a constant risk premium, c. If equation (8.77) is considered, it can be seen that by subtracting from both sides of the relationship we have (8.78) Examination of equation (8.78) generates some interesting restrictions. If the interest rates under analysis, say and are I(1) series, then, by definition, and will be stationary series. There is a general acceptance that interest rates, Treasury bill yields, etc. are well described as I(1) processes and this can be seen in Campbell and Shiller (1988) and Stock and Watson (1988). Further, since c is a constant then it is by definition a stationary series. Consequently, if the EH is to hold, given that c and are I(0) implying that the RHS of equation (8.78) is stationary, then must by definition be stationary, otherwise we will have an inconsistency in the order of integration between the RHS and LHS of the relationship. is commonly known as the spread between the nperiod and m-period rates, denoted which in turn gives an indication of the slope of the term structure. Consequently, it follows that if the EH is 489

to hold, then the spread will be found to be stationary and therefore and will cointegrate with a cointegrating vector (1, −1) for Therefore, the integrated process driving each of the two rates is common to both and hence it can be said that the rates have a common stochastic trend. As a result, since the EH predicts that each interest rate series will cointegrate with the one-period interest rate, it must be true that the stochastic process driving all the rates is the same as that driving the oneperiod rate, i.e., any combination of rates formed to create a spread should be found to cointegrate with a cointegrating vector (1, −1). Many examinations of the expectations hypothesis of the term structure have been conducted in the literature, and still no overall consensus appears to have emerged concerning its validity. One such study that tested the expectations hypothesis using a standard data set due to McCulloch (1987) was conducted by Shea (1992). The data comprises a zero coupon term structure for various maturities from one month to twenty-five years, covering the period January 1952–February 1987. Various techniques are employed in Shea’s paper, while only his application of the Johansen technique is discussed here. A vector Xt containing the interest rate at each of the maturities is constructed (8.79) where Rt denotes the spot interest rate. It is argued that each of the elements of this vector is non-stationary, and hence the Johansen approach is used to model the system of interest rates and to test for cointegration between the rates. Both the λmax and λtrace statistics are employed, corresponding to the use of the maximum eigenvalue and the cumulated eigenvalues, respectively. Shea tests for cointegration between various combinations of the interest rates, measured as returns to maturity. A selection of Shea’s results is presented in Table 8.14. Table 8.14 Tests of the expectations hypothesis using the US zero coupon yield curve with monthly data

Sample period 1952M1–

Interest rates included

Lag length of VAR

Hypothesis is

λmax

λtrace

2

r=0

47.54***

49.82***

490

1978M12 1952M1– 1987M2 1952M1– 1987M2

2

2

r≤1

2.28

2.28

r=0

40.66***

43.73***

r≤1

3.07

3.07

r=0

40.13***

42.63***

r≤1

2.50

2.50

r =0

34.78***

75.50***

r ≤1

23.31*

40.72

r ≤2

11.94

17.41

r ≤3

3.80

5.47

r ≤4

1.66

1.66

1973M5– 1987M2 7

Notes:*,** and*** denote significance at the 20%, 10% and 5% levels, respectively; r is the number of cointegrating vectors under the null hypothesis. Source: Shea (1992). Reprinted with the permission of American Statistical Association. All rights reserved.

The results below, together with the other results presented by Shea, seem to suggest that the interest rates at different maturities are typically cointegrated, usually with one cointegrating vector. As one may have expected, the cointegration becomes weaker in the cases where the analysis involves rates a long way apart on the maturity spectrum. However, cointegration between the rates is a necessary but not sufficient condition for the expectations hypothesis of the term structure to be vindicated by the data. Validity of the expectations hypothesis also requires that any combination of rates formed to create a spread should be found to cointegrate with a cointegrating vector (1, −1). When comparable restrictions are placed on the β estimates associated with the cointegrating vectors, they are typically rejected, suggesting only limited support for the expectations hypothesis. A Note on Long-memory Models 491

It is widely believed that (the logs of) asset prices contain a unit root. However, asset return series evidently do not possess a further unit root, although this does not imply that the returns are independent. In particular, it is possible (and indeed, it has been found to be the case with some financial and economic data) that observations from a given series taken some distance apart, show signs of dependence. Such series are argued to possess long memory. One way to represent this phenomenon is using a ‘fractionally integrated’ model. In simple terms, a series is integrated of a given order d if it becomes stationary on differencing a minimum of d times. In the fractionally integrated framework, d is allowed to take on non-integer values. This framework has been applied to the estimation of ARMA models (see, for example, Mills, 2008). Under fractionally integrated models, the corresponding autocorrelation function (ACF) will decline hyperbolically, rather than exponentially to zero. Thus, the ACF for a fractionally integrated model dies away considerably more slowly than that of an ARMA model with d = 0. The notion of long memory has also been applied to GARCH models (discussed in Chapter 9), where volatility has been found to exhibit longrange dependence. A new class of models known as fractionally integrated GARCH (FIGARCH) have been proposed to allow for this phenomenon (see Ding, Granger, and Engle, 1993 or Bollerslev and Mikkelsen, 1996). KEY CONCEPTS The key terms to be able to define and explain from this chapter are non-stationary unit root augmented Dickey–Fuller test error correction model Johansen technique eigenvalues explosive process spurious regression cointegration Engle–Granger 2-step approach vector error correction model

SELF-STUDY QUESTIONS 492

1. (a) What kinds of variables are likely to be non-stationary? How can such variables be made stationary? (b) Why is it in general important to test for non-stationarity in time series data before attempting to build an empirical model? (c) Define the following terms and describe the processes that they represent (i) Weak stationarity (ii) Strict stationarity (iii) Deterministic trend (iv) Stochastic trend. 2. A researcher wants to test the order of integration of some timeseries data. He decides to use the DF test. He estimates a regression of the form and obtains the estimate with standard error = 0.31. (a) What are the null and alternative hypotheses for this test? (b) Given the data, and a critical value of −2.88, perform the test. (c) What is the conclusion from this test and what should be the next step? (d) Why is it not valid to compare the estimated test statistic with the corresponding critical value from a t-distribution, even though the test statistic takes the form of the usual tratio? 3. Using the same regression as for Question 2, but on a different set of data, the researcher now obtains the estimate with standard error = 0.16. (a) Perform the test. (b) What is the conclusion, and what should be the next step? (c) Another researcher suggests that there may be a problem with this methodology since it assumes that the disturbances (ut) are white noise. Suggest a possible source of difficulty and how the researcher might in practice get around it. 4. (a) Consider a series of values for the spot and futures prices of a 493

given commodity. In the context of these series, explain the concept of cointegration. Discuss how a researcher might test for cointegration between the variables using the Engle– Granger approach. Explain also the steps involved in the formulation of an error correction model. (b) Give a further example from finance where cointegration between a set of variables may be expected. Explain, by reference to the implication of non-cointegration, why cointegration between the series might be expected. 5. (a) Briefly outline Johansen’s methodology for testing for cointegration between a set of variables in the context of a VAR. (b) A researcher uses the Johansen procedure and obtains the following test statistics (and critical values) r

λmax

0 1 2 3 4

38.962 29.148 16.304 8.861 1.994

5% critical value 33.178 27.169 20.278 14.036 3.962

Determine the number of cointegrating vectors. (c) ‘If two series are cointegrated, it is not possible to make inferences regarding the cointegrating relationship using the Engle–Granger technique since the residuals from the cointegrating regression are likely to be autocorrelated.’ How does Johansen circumvent this problem to test hypotheses about the cointegrating relationship? (d) Give one or more examples from the academic finance literature of where the Johansen systems technique has been employed. What were the main results and conclusions of this research? (e) Compare the Johansen maximal eigenvalue test with the test based on the trace statistic. State clearly the null and alternative hypotheses in each case. 494

6. (a) Suppose that a researcher has a set of three variables, yt (t = 1, …, T), i.e., yt denotes a p-variate, or p × 1 vector, that she wishes to test for the existence of cointegrating relationships using the Johansen procedure. What is the implication of finding that the rank of the appropriate matrix takes on a value of (i) 0 (ii) 1 (iii) 2 (iv) 3? (b) The researcher obtains results for the Johansen test using the variables outlined in part (a) of the question as follows r

λmax

0 1 2 3

38.65 26.91 10.67 8.55

5% critical value 30.26 23.84 17.72 10.71

Determine the number of cointegrating vectors, explaining your answer. 7. Compare and contrast the Engle–Granger and Johansen methodologies for testing for cointegration and modelling cointegrated systems. Which, in your view, represents the superior approach and why? 8. (a) What issues arise when testing for a unit root if there is a structural break in the series under investigation? (b) What are the limitations of the Perron (1989) approach for dealing with structural breaks in testing for a unit root?

1

2 3

This material is fairly specialised and thus is not well covered by most of the standard textbooks. But for any readers wishing to see more detail, there is a useful and accessible chapter by Perron in Rao (1994). There is also a chapter on structural change in Maddala and Kim (1999). EuroSterling interest rates are those at which money is loaned/borrowed in British pounds but outside of the UK. For further reading on this topic, the book by Harris (1995) provides an extremely clear introduction to unit roots and cointegration, including a section on seasonal unit roots.

495

2

Strictly, the eigenvalues used in the test statistics are taken from rankrestricted product moment matrices and not of Π itself.

496

9 Modelling Volatility and Correlation

LEARNING OUTCOMES In this chapter, you will learn how to Discuss the features of data that motivate the use of GARCH models Explain how conditional volatility models are estimated Test for ‘ARCH-effects’ in time-series data Produce forecasts from GARCH models Contrast various models from the GARCH family Discuss the three hypothesis testing procedures available under maximum likelihood estimation Construct multivariate conditional volatility models and compare between alternative specifications

9.1 Motivations: An Excursion into Non-Linearity Land All of the models that have been discussed in Chapters 3–8 of this book have been linear in nature – that is, the model is linear in the parameters, so that there is one parameter multiplied by each variable in the model. For example, a structural model could be something like (9.1) or more compactly y = Xβ + u. It was additionally assumed that ut ~ N(0, σ2). 497

The linear paradigm as described above is a useful one. The properties of linear estimators are very well researched and very well understood. Many models that appear, prima facie, to be non-linear, can be made linear by taking logarithms or some other suitable transformation. However, it is likely that many relationships in finance are intrinsically non-linear. As Campbell, Lo and MacKinlay (1997) state, the payoffs to options are nonlinear in some of the input variables, and investors’ willingness to trade off returns and risks are also non-linear. These observations provide clear motivations for consideration of non-linear models in a variety of circumstances in order to capture better the relevant features of the data. Linear structural (and time series) models such as equation 9.1 are also unable to explain a number of important features common to much financial data, including Leptokurtosis – that is, the tendency for financial asset returns to have distributions that exhibit fat tails and excess peakedness at the mean. Volatility clustering or volatility pooling – the tendency for volatility in financial markets to appear in bunches. Thus large returns (of either sign) are expected to follow large returns, and small returns (of either sign) to follow small returns. A plausible explanation for this phenomenon, which seems to be an almost universal feature of asset return series in finance, is that the information arrivals which drive price changes themselves occur in bunches rather than being evenly spaced over time. Leverage effects – the tendency for volatility to rise more following a large price fall than following a price rise of the same magnitude. Campbell et al. (1997) broadly define a non-linear data generating process as one where the current value of the series is related non-linearly to current and previous values of the error term (9.2) where ut is an iid error term and f is a non-linear function. According to Campbell et al., a more workable and slightly more specific definition of a non-linear model is given by the equation (9.3) where g is a function of past error terms only, and σ2 can be interpreted as 498

a variance term, since it is multiplied by the current value of the error. Campbell et al. usefully characterise models with non-linear g(•) as being non-linear in mean, while those with non-linear σ(•)2 are characterised as being non-linear in variance. Models can be linear in mean and variance (e.g., the CLRM, ARMA models) or linear in mean, but non-linear in variance (e.g., GARCH models). Models could also be classified as non-linear in mean but linear in variance (e.g., bicorrelations models, a simple example of which is of the following form (see Brooks and Heravi, 1999)) (9.4) Finally, models can be non-linear in both mean and variance (e.g., the hybrid threshold model with GARCH errors employed by Brooks, 2001).

9.1.1 Types of Non-Linear Models There are an infinite number of different types of non-linear model. However, only a small number of non-linear models have been found to be useful for modelling financial data. The most popular non-linear financial models are the ARCH or GARCH models used for modelling and forecasting volatility, and switching models, which allow the behaviour of a series to follow different processes at different points in time. Models for volatility and correlation will be discussed in this chapter, with switching models being covered in Chapter 10.

9.1.2 Testing for Non-Linearity How can it be determined whether a non-linear model may potentially be appropriate for the data? The answer to this question should come at least in part from financial theory: a non-linear model should be used where financial theory suggests that the relationship between variables should be such as to require a non-linear model. But the linear versus non-linear choice may also be made partly on statistical grounds – deciding whether a linear specification is sufficient to describe all of the most important features of the data at hand. So what tools are available to detect non-linear behaviour in financial time series? Unfortunately, ‘traditional’ tools of time-series analysis (such as estimates of the autocorrelation or partial autocorrelation function, or ‘spectral analysis’, which involves looking at the data in the frequency 499

domain) are likely to be of little use. Such tools may find no evidence of linear structure in the data, but this would not necessarily imply that the same observations are independent of one another. However, there are a number of tests for non-linear patterns in time series that are available to the researcher. These tests can broadly be split into two types: general tests and specific tests. General tests, also sometimes called ‘portmanteau’ tests, are usually designed to detect many departures from randomness in data. The implication is that such tests will detect a variety of non-linear structures in data, although these tests are unlikely to tell the researcher which type of non-linearity is present! Perhaps the simplest general test for non-linearity is Ramsey’s RESET test discussed in Chapter 4, although there are many other popular tests available. One of the most widely used tests is known as the BDS test (see Brock, Hsieh and LeBaron, 1991) named after the three authors who first developed it. BDS is a pure hypothesis test. That is, it has as its null hypothesis that the data are pure noise (completely random), and it has been argued to have power to detect a variety of departures from randomness – linear or non-linear stochastic processes, deterministic chaos, etc. (see Brock et al., 1991). The BDS test follows a standard normal distribution under the null hypothesis. The details of this test, and others, are technical and beyond the scope of this book, although computer code for BDS estimation is now widely available free of charge on the internet. As well as applying the BDS test to raw data in an attempt to ‘see if there is anything there’, another suggested use of the test is as a model diagnostic. The idea is that a proposed model (e.g., a linear model, GARCH, or some other non-linear model) is estimated, and the test applied to the (standardised) residuals in order to ‘see what is left’. If the proposed model is adequate, the standardised residuals should be white noise, while if the postulated model is insufficient to capture all of the relevant features of the data, the BDS test statistic for the standardised residuals will be statistically significant. This is an excellent idea in theory, but has difficulties in practice. First, if the postulated model is a non-linear one (such as GARCH), the asymptotic distribution of the test statistic will be altered, so that it will no longer follow a normal distribution. This requires new critical values to be constructed via simulation for every type of non-linear model whose residuals are to be tested. More seriously, if a non-linear model is fitted to the data, any remaining structure is typically garbled, resulting in the test either being unable to detect additional structure present in the data (see Brooks and Henry, 2000) or selecting as 500

adequate a model which is not even in the correct class for that data generating process (see Brooks and Heravi, 1999). Other popular tests for non-linear structure in time-series data include the bispectrum test due to Hinich (1982), the bicorrelation test (see Hsieh, 1993; Hinich, 1996; or Brooks and Heravi, 1999 for its multivariate generalisation). Most applications of the above tests conclude that there is non-linear dependence in financial asset returns series, but that the dependence is best characterised by a GARCH-type process (see Hinich and Patterson, 1985; Baillie and Bollerslev, 1989; Brooks, 1996; and the references therein for applications of non-linearity tests to financial data). Specific tests, on the other hand, are usually designed to have power to find specific types of non-linear structure. Specific tests are unlikely to detect other forms of nonlinearities in the data, but their results will by definition offer a class of models that should be relevant for the data at hand. Examples of specific tests will be offered later in this and subsequent chapters.

9.1.3 Chaos in Financial Markets Econometricians have searched long and hard for chaos in financial, macroeconomic and microeconomic data, with very limited success to date. Chaos theory is a notion taken from the physical sciences that suggests that there could be a deterministic, non-linear set of equations underlying the behaviour of financial series or markets. Such behaviour will appear completely random to the standard statistical tests developed for application to linear models. The motivation behind this endeavour is clear: a positive sighting of chaos implies that while, by definition, longterm forecasting would be futile, short-term forecastability and controllability are possible, at least in theory, since there is some deterministic structure underlying the data. Varying definitions of what actually constitutes chaos can be found in the literature, but a robust definition is that a system is chaotic if it exhibits sensitive dependence on initial conditions (SDIC). The concept of SDIC embodies the fundamental characteristic of chaotic systems that if an infinitesimal change is made to the initial conditions (the initial state of the system), then the corresponding change iterated through the system for some arbitrary length of time will grow exponentially. Although several statistics are commonly used to test for the presence of chaos, only one is arguably a true test for chaos, namely estimation of the largest Lyapunov exponent. 501

The largest Lyapunov exponent measures the rate at which information is lost from a system. A positive largest Lyapunov exponent implies sensitive dependence, and therefore that evidence of chaos has been obtained. This has important implications for the predictability of the underlying system, since the fact that all initial conditions are in practice estimated with some error (owing either to measurement error or exogenous noise), will imply that long-term forecasting of the system is impossible as all useful information is likely to be lost in just a few time steps. Chaos theory was hyped and embraced by both the academic literature and in financial markets worldwide in the 1980s. However, almost without exception, applications of chaos theory to financial markets have been unsuccessful. Consequently, although the ideas generate continued debate owing to the interesting mathematical properties and the possibility of finding a prediction holy grail, academic and practitioner interest in chaotic models for financial markets has arguably almost disappeared. The primary reason for the failure of the chaos theory approach appears to be the fact that financial markets are extremely complex, involving a very large number of different participants, each with different objectives and different sets of information – and, above all, each of whom are human with human emotions and irrationalities. The consequence of this is that financial and economic data are usually far noisier and ‘more random’ than data from other disciplines, making the specification of a deterministic model very much harder and possibly even futile.

9.1.4 Neural Network Models Artificial neural networks (ANNs) are a class of models whose structure is broadly motivated by the way that the brain performs computation. ANNs have been widely employed in finance for tackling time-series and classification problems. Recent applications have included forecasting financial asset returns, volatility, bankruptcy and takeover prediction. Applications are contained in the books by Trippi and Turban (1993), Van Eyden (1996) and Refenes (1995). A technical collection of papers on the econometric aspects of neural networks is given by White (1992), while an excellent general introduction and a description of the issues surrounding neural network model estimation and analysis is contained in Franses and van Dijk (2000). Neural networks have virtually no theoretical motivation in finance (they are often termed a ‘black box’ technology), but owe their popularity to their ability to fit any functional relationship in the data to an arbitrary 502

degree of accuracy. The most common class of ANN models in finance are known as feedforward network models. These have a set of inputs (akin to regressors) linked to one or more outputs (akin to the regressand) via one or more ‘hidden’ or intermediate layers. The size and number of hidden layers can be modified to give a closer or less close fit to the data sample, while a feedforward network with no hidden layers is simply a standard linear regression model. Neural network models are likely to work best in situations where financial theory has virtually nothing to say about the likely functional form for the relationship between a set of variables. However, their popularity has arguably waned over the past five years or so as a consequence of several perceived problems with their employment. First, the coefficient estimates from neural networks do not have any real theoretical interpretation. Second, virtually no diagnostic or specification tests are available for estimated models to determine whether the model under consideration is adequate. Third, ANN models can provide excellent fits in-sample to a given set of ‘training’ data, but typically provide poor out-of-sample forecast accuracy. The latter result usually arises from the tendency of neural networks to fit closely to sample-specific data features and ‘noise’, and therefore their inability to generalise. Various methods of resolving this problem exist, including ‘pruning’ (removing some parts of the network) or the use of information criteria to guide the network size. Finally, the non-linear estimation of neural network models can be cumbersome and computationally time-intensive, particularly, for example, if the model must be estimated rolling through a sample to produce a series of one-step-ahead forecasts.

9.2 Models for Volatility Modelling and forecasting stock market volatility has been the subject of vast empirical and theoretical investigation over the past decade or so by academics and practitioners alike. There are a number of motivations for this line of inquiry. Arguably, volatility is one of the most important concepts in the whole of finance. Volatility, as measured by the standard deviation or variance of returns, is often used as a crude measure of the total risk of financial assets. Many value-at-risk models for measuring market risk require the estimation or forecast of a volatility parameter. The volatility of stock market prices also enters directly into the Black–Scholes formula for deriving the prices of traded options. The next few sections will discuss various models that are appropriate to 503

capture the stylised features of volatility, discussed below, that have been observed in the literature.

9.3 Historical Volatility The simplest model for volatility is the historical estimate. Historical volatility simply involves calculating the variance (or standard deviation) of returns in the usual way over some historical period, and this then becomes the volatility forecast for all future periods. The historical average variance (or standard deviation) was traditionally used as the volatility input to options pricing models, although there is a growing body of evidence suggesting that the use of volatility predicted from more sophisticated time-series models will lead to more accurate option valuations (see, for example, Akgiray, 1989; or Chu and Freund, 1996). Historical volatility is still useful as a benchmark for comparing the forecasting ability of more complex time models.

9.4 Implied Volatility Models All pricing models for financial options require a volatility estimate or forecast as an input. Given the price of a traded option obtained from transactions data, it is possible to determine the volatility forecast over the lifetime of the option implied by the option’s valuation. For example, if the standard Black–Scholes model is used, the option price, the time to maturity, a risk-free rate of interest, the strike price and the current value of the underlying asset, are all either specified in the details of the options contracts or are available from market data. Therefore, given all of these quantities, it is possible to use a numerical procedure, such as the method of bisections or Newton–Raphson to derive the volatility implied by the option (see Watsham and Parramore, 2004). This implied volatility is the market’s forecast of the volatility of underlying asset returns over the lifetime of the option.

9.5 Exponentially Weighted Moving Average Models The exponentially weighted moving average (EWMA) is essentially a simple extension of the historical average volatility measure, which allows more recent observations to have a stronger impact on the forecast of volatility than older data points. Under an EWMA specification, the latest observation carries the largest weight, and weights associated with 504

previous observations decline exponentially over time. This approach has two advantages over the simple historical model. First, volatility is in practice likely to be affected more by recent events, which carry more weight, than events further in the past. Second, the effect on volatility of a single given observation declines at an exponential rate as weights attached to recent events fall. On the other hand, the simple historical approach could lead to an abrupt change in volatility once the shock falls out of the measurement sample. And if the shock is still included in a relatively long measurement sample period, then an abnormally large observation will imply that the forecast will remain at an artificially high level even if the market is subsequently tranquil. The exponentially weighted moving average model can be expressed in several ways, e.g., (9.5) where is the estimate of the variance for period t, which also becomes the forecast of future volatility for all periods, is the average return estimated over the observations and λ is the ‘decay factor’, which determines how much weight is given to recent versus older observations. The decay factor could be estimated, but in many studies is set at 0.94 as recommended by RiskMetrics, producers of popular risk measurement software. Note also that RiskMetrics and many academic papers assume that the average return, is zero. For data that is of daily frequency or higher, this is not an unreasonable assumption, and is likely to lead to negligible loss of accuracy since it will typically be very small. Obviously, in practice, an infinite number of observations will not be available on the series, so that the sum in equation (9.5) must be truncated at some fixed lag. As with exponential smoothing models, the forecast from an EWMA model for all prediction horizons is the most recent weighted average estimate. It is worth noting two important limitations of EWMA models. First, while there are several methods that could be used to compute the EWMA, the crucial element in each case is to remember that when the infinite sum in equation (9.5) is replaced with a finite sum of observable data, the weights from the given expression will now sum to less than one. In the case of small samples, this could make a large difference to the computed EWMA and thus a correction may be necessary. Second, most time series models, such as GARCH (see below), will have forecasts that tend towards the unconditional variance of the series as the prediction horizon increases. 505

This is a good property for a volatility forecasting model to have, since it is well known that volatility series are ‘mean-reverting’. This implies that if they are currently at a high level relative to their historic average, they will have a tendency to fall back towards their average level, while if they are at a low level relative to their historic average, they will have a tendency to rise back towards the average. This feature is accounted for in GARCH volatility forecasting models, but not by EWMAs.

9.6 Autoregressive Volatility Models Autoregressive volatility models are a relatively simple example from the class of stochastic volatility specifications. The idea is that a time series of observations on some volatility proxy are obtained. The standard Box– Jenkins-type procedures for estimating autoregressive (or ARMA) models can then be applied to this series. If the quantity of interest in the study is a daily volatility estimate, two natural proxies have been employed in the literature: squared daily returns, or daily range estimators. Producing a series of daily squared returns trivially involves taking a column of observed returns and squaring each observation. The squared return at each point in time, t, then becomes the daily volatility estimate for day t. A range estimator typically involves calculating the log of the ratio of the highest observed price to the lowest observed price for trading day t, which then becomes the volatility estimate for day t (9.6) Given either the squared daily return or the range estimator, a standard autoregressive model is estimated, with the coefficients βi estimated using OLS (or maximum likelihood – see below). The forecasts are also produced in the usual fashion discussed in Chapter 6 in the context of ARMA models (9.7)

9.7 Autoregressive Conditionally Heteroscedastic (ARCH) Models

506

One particular non-linear model in widespread usage in finance is known as an ‘ARCH’ model (ARCH stands for ‘autoregressive conditionally heteroscedastic’). To see why this class of models is useful, recall that a typical structural model could be expressed by an equation of the form given in equation (9.1) on p. 384 with ut ~ N(0, σ2). The assumption of the CLRM that the variance of the errors is constant is known as homoscedasticity (i.e., it is assumed that var(ut) = σ2). If the variance of the errors is not constant, this would be known as heteroscedasticity. As was explained in Chapter 5, if the errors are heteroscedastic, but assumed homoscedastic, an implication would be that standard error estimates could be wrong. It is unlikely in the context of financial time series that the variance of the errors will be constant over time, and hence it makes sense to consider a model that does not assume that the variance is constant, and which describes how the variance of the errors evolves. Another important feature of many series of financial asset returns that provides a motivation for the ARCH class of models, is known as ‘volatility clustering’ or ‘volatility pooling’. Volatility clustering describes the tendency of large changes in asset prices (of either sign) to follow large changes and small changes (of either sign) to follow small changes. In other words, the current level of volatility tends to be positively correlated with its level during the immediately preceding periods. This phenomenon is demonstrated in Figure 9.1, which plots daily S&P500 returns for August 2003–July 2018.

507

Figure 9.1 Daily S&P returns for August 2003–July 2018

The important point to note from Figure 9.1 is that volatility occurs in bursts. There appears to have been a prolonged period of relative tranquillity in the market during the 2003 to 2008 period until the financial crisis began, evidenced by only relatively small positive and negative returns until that point. On the other hand, during mid-2008 to mid-2009, there was far more volatility, when many large positive and large negative returns were observed during a short space of time. Abusing the terminology slightly, it could be stated that ‘volatility is autocorrelated’. How could this phenomenon, which is common to many series of financial asset returns, be parameterised (modelled)? One approach is to use an ARCH model. To understand how the model works, a definition of the conditional variance of a random variable, ut, is required. The distinction between the conditional and unconditional variances of a random variable is exactly the same as that of the conditional and unconditional mean. The conditional variance of ut may be denoted which is written as (9.8) It is usually assumed that E(ut) = 0, so (9.9) Equation (9.9) states that the conditional variance of a zero mean normally distributed random variable ut is equal to the conditional expected value of the square of ut. Under the ARCH model, the ‘autocorrelation in volatility’ is modelled by allowing the conditional variance of the error term, to depend on the immediately previous value of the squared error (9.10) The above model is known as an ARCH(1), since the conditional variance depends on only one lagged squared error. Notice that equation (9.10) is only a partial model, since nothing has been said yet about the conditional mean. Under ARCH, the conditional mean equation (which describes how the dependent variable, yt, varies over time) could take almost any form that the researcher wishes. One example of a full model would be 508

(9.11) (9.12) The model given by equations (9.11) and (9.12) could easily be extended to the general case where the error variance depends on q lags of squared errors, which would be known as an ARCH(q) model: (9.13) Instead of calling the conditional variance called ht, so that the model would be written

in the literature it is often

(9.14) (9.15) The remainder of this chapter will use to denote the conditional variance at time t, except for computer instructions where ht will be used since it is easier not to use Greek letters.

9.7.1 Another Way of Expressing ARCH Models For illustration, consider an ARCH(1). The model can be expressed in two ways that look different but are in fact identical. The first is as given in equations (9.11) and (9.12) above. The second way would be as follows (9.16) (9.17) (9.18) The form of the model given in equations (9.11) and (9.12) is more commonly presented, although specifying the model as in equations (9.16)–(9.18) is required in order to use a GARCH process in a simulation study (see Chapter 13). To show that the two methods for expressing the model are equivalent, consider that in equation (9.17), vt is normally distributed with zero mean and unit variance, so that ut will also be normally distributed with zero mean and variance

9.7.2 Non-Negativity Constraints 509

Since ht is a conditional variance, its value must always be strictly positive; a negative variance at any point in time would be meaningless. The variables on the RHS of the conditional variance equation are all squares of lagged errors, and so by definition will not be negative. In order to ensure that these always result in positive conditional variance estimates, all of the coefficients in the conditional variance are usually required to be non-negative. If one or more of the coefficients were to take on a negative value, then for a sufficiently large lagged squared innovation term attached to that coefficient, the fitted value from the model for the conditional variance could be negative. This would clearly be nonsensical. So, for example, in the case of equation (9.18), the non-negativity condition would be α0 ≥ 0 and α1 ≥ 0. More generally, for an ARCH(q) model, all coefficients would be required to be non-negative: αi ≥ 0 ∀ i = 0, 1, 2, …, q. In fact, this is a sufficient but not necessary condition for non-negativity of the conditional variance (i.e., it is a slightly stronger condition than is actually necessary).

9.7.3 Testing for ‘ARCH Effects’ A test for determining whether ‘ARCH effects’ are present in the residuals of an estimated model may be conducted using the steps outlined in Box 9.1. BOX 9.1 Testing for ‘ARCH effects’ (1) Run any postulated linear regression of the form given in the equation above, e.g., (9.19) saving the residuals, (2) Square the residuals, and regress them on q own lags to test for ARCH of order q, i.e., run the regression (9.20) where vt is an error term. Obtain R2 from this regression. (3) The test statistic is defined as TR2 (the number of observations 510

multiplied by the coefficient of multiple correlation) from the last regression, and is distributed as a χ2(q) (4) The null and alternative hypotheses are

Thus, the test is one of a joint null hypothesis that all q lags of the squared residuals have coefficient values that are not significantly different from zero. If the value of the test statistic is greater than the critical value from the χ2 distribution, then reject the null hypothesis. The test can also be thought of as a test for autocorrelation in the squared residuals. As well as testing the residuals of an estimated model, the ARCH test is frequently applied to raw returns data.

9.7.4 Limitations of ARCH(q) Models ARCH provided a framework for the analysis and development of time series models of volatility. However, ARCH models themselves have rarely been used in the last decade or more, since they bring with them a number of difficulties How should the value of q, the number of lags of the squared residual in the model, be decided? One approach to this problem would be the use of a likelihood ratio test, discussed later in this chapter, although there is no clearly best approach. The value of q, the number of lags of the squared error that are required to capture all of the dependence in the conditional variance, might be very large. This would result in a large conditional variance model that was not parsimonious. Engle (1982) circumvented this problem by specifying an arbitrary linearly declining lag length on an ARCH(4) (9.21) such that only two parameters are required in the conditional variance equation (γ0 and γ1), rather than the five which would be required for an unrestricted ARCH(4). Non-negativity constraints might be violated. Everything else equal, 511

the more parameters there are in the conditional variance equation, the more likely it is that one or more of them will have negative estimated values. A natural extension of an ARCH(q) model which overcomes some of these problems is a GARCH model. In contrast with ARCH, GARCH models are extremely widely employed in practice.

9.8 Generalised ARCH (GARCH) Models The GARCH model was developed independently by Bollerslev (1986) and Taylor (1986). The GARCH model allows the conditional variance to be dependent upon previous own lags, so that the conditional variance equation in the simplest case is now (9.22) This is a GARCH(1,1) model. is known as the conditional variance since it is a one-period ahead estimate for the variance calculated based on any past information thought relevant. Using the GARCH model it is possible to interpret the current fitted variance, ht, as a weighted function of a long-term average value (dependent on α0), information about volatility during the previous period and the fitted variance from the model during the previous period Note that the GARCH model can be expressed in a form that shows that it is effectively an ARMA model for the conditional variance. To see this, consider that the squared return at time t relative to the conditional variance is given by (9.23) or (9.24) Using the latter expression to substitute in for the conditional variance in equation (9.22) (9.25) Rearranging 512

(9.26) so that (9.27) This final expression is an ARMA(1,1) process for the squared errors. Why is GARCH a better and therefore a far more widely used model than ARCH? The answer is that the former is more parsimonious, and avoids overfitting. Consequently, the model is less likely to breach nonnegativity constraints. In order to illustrate why the model is parsimonious, first take the conditional variance equation in the GARCH(1,1) case, subtract 1 from each of the time subscripts of the conditional variance equation in equation (9.22), so that the following expression would be obtained (9.28) and subtracting 1 from each of the time subscripts again (9.29) Substituting into equation (9.22) for (9.30) (9.31) Now substituting into equation (9.31) for (9.32) (9.33) (9.34) An infinite number of successive substitutions of this kind would yield (9.35) The first expression on the RHS of equation (9.35) is simply a constant, and as the number of observations tends to infinity, β∞ will tend to zero. Hence, the GARCH(1,1) model can be written as 513

(9.36) (9.37) which is a restricted infinite order ARCH model. Thus the GARCH(1,1) model, containing only three parameters in the conditional variance equation, is a very parsimonious model, that allows an infinite number of past squared errors to influence the current conditional variance. The GARCH(1,1) model can be extended to a GARCH(p,q) formulation, where the current conditional variance is parameterised to depend upon q lags of the squared error and p lags of the conditional variance (9.38) (9.39) But in general a GARCH(1,1) model will be sufficient to capture the volatility clustering in the data, and rarely is any higher order model estimated or even entertained in the academic finance literature.

9.8.1 The Unconditional Variance Under a GARCH Specification The conditional variance is changing, but the unconditional variance of ut is constant and given by (9.40) so long as α1 + β < 1. For α1 + β ≥ 1, the unconditional variance of ut is not defined, and this would be termed ‘non-stationarity in variance’. α1 + β = 1 would be known as a ‘unit root in variance’, also termed ‘Integrated GARCH’ or IGARCH. Non-stationarity in variance does not have a strong theoretical motivation for its existence, as would be the case for nonstationarity in the mean (e.g., of a price series). Furthermore, a GARCH model whose coefficients imply non-stationarity in variance would have some highly undesirable properties. One illustration of these relates to the forecasts of variance made from such models. For stationary GARCH 514

models, conditional variance forecasts converge upon the long-term average value of the variance as the prediction horizon increases (see below). For IGARCH processes, this convergence will not happen, while for α1 + β > 1, the conditional variance forecast will tend to infinity as the forecast horizon increases.

9.9 Estimation of ARCH/GARCH Models Since the model is no longer of the usual linear form, OLS cannot be used for GARCH model estimation. There are a variety of reasons for this, but the simplest and most fundamental is that OLS minimises the RSS. The RSS depends only on the parameters in the conditional mean equation, and not the conditional variance, and hence RSS minimisation is no longer an appropriate objective. In order to estimate models from the GARCH family, another technique known as maximum likelihood is employed. Essentially, the method works by finding the most likely values of the parameters given the actual data. More specifically, a log-likelihood function (LLF) is formed and the values of the parameters that maximise it are sought. Maximum likelihood estimation can be employed to find parameter values for both linear and non-linear models. The steps involved in actually estimating an ARCH or GARCH model are shown in Box 9.2. The following section will elaborate on points (2) and (3) presented in the box, explaining how the LLF is derived. BOX 9.2 Estimating an ARCH or GARCH model (1) Specify the appropriate equations for the mean and the variance – e.g. an AR(1)-GARCH(1,1) model (9.41) (9.42) (2) Specify the log-likelihood function (LLF) to maximise under a normality assumption for the disturbances (9.43)

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(3) The computer will maximise the function and generate parameter values that maximise the LLF and will construct their standard errors.

9.9.1 Parameter Estimation Using Maximum Likelihood As stated above, under maximum likelihood estimation, a set of parameter values are chosen that are most likely to have produced the observed data. This is done by first forming a likelihood function, denoted LF. LF will be a multiplicative function of the actual data, which will consequently be difficult to maximise with respect to the parameters. Therefore, its logarithm is taken in order to turn LF into an additive function of the sample data, i.e., the LLF. A derivation of the maximum likelihood (ML) estimator in the context of the simple bivariate regression model with homoscedasticity is given in the appendix to this chapter. Essentially, deriving the ML estimators involves differentiating the LLF with respect to the parameters. But how does this help in estimating heteroscedastic models? How can the method outlined in Appendix 9.1 to this chapter for homoscedastic models be modified for application to GARCH model estimation? In the context of conditional heteroscedasticity models, the model is so that the variance of the errors has been modified from being assumed constant, σ2, to being time-varying, with the equation for the conditional variance as previously. The LLF relevant for a GARCH model can be constructed in the same way as for the homoscedastic case by replacing

with the equivalent for time-varying variance

and replacing σ2 in the denominator of the last part of the expression with (see Appendix 9.1 to this chapter). Derivation of this result from first principles is beyond the scope of this text, but the log-likelihood function for the above model with time-varying conditional variance and normally distributed errors is given by equation (9.43) in Box 9.2. Intuitively, maximising the LLF involves jointly minimising 516

and

(since these terms appear preceded with a negative sign in the LLF, and

is just a constant with respect to the parameters). Minimising these terms jointly also implies minimising the error variance, as described in Chapter 4. Unfortunately, maximising the LLF for a model with time-varying variances is trickier than in the homoscedastic case. Analytical derivatives of the LLF in equation (9.43) with respect to the parameters have been developed, but only in the context of the simplest examples of GARCH specifications. Moreover, the resulting formulae are complex, so a numerical procedure is often used instead to maximise the log-likelihood function. Essentially, all methods work by ‘searching’ over the parameter-space until the values of the parameters that maximise the log-likelihood function are found. Most software packages employ an iterative technique for maximising the LLF. This means that, given a set of initial guesses for the parameter estimates, these parameter values are updated at each iteration until the program determines that an optimum has been reached. If the LLF has only one maximum with respect to the parameter values, any optimisation method should be able to find it – although some methods will take longer than others. A detailed presentation of the various methods available is beyond the scope of this book. However, as is often the case with non-linear models such as GARCH, the LLF can have many local maxima, so that different algorithms could find different local maxima of the LLF. Hence readers should be warned that different optimisation procedures could lead to different coefficient estimates and especially different estimates of the standard errors (see Brooks, 2001 or 2003 for details). In such instances, a good set of initial parameter guesses is essential. Local optima or multimodalities in the likelihood surface present potentially serious drawbacks with the maximum likelihood approach to estimating the parameters of a GARCH model, as shown in Figure 9.2. 517

Figure 9.2 The problem of local optima in maximum likelihood

estimation Suppose that the model contains only one parameter, θ, so that the loglikelihood function is to be maximised with respect to this one parameter. In Figure 9.2, the value of the LLF for each value of θ is denoted l(θ). Clearly, l(θ) reaches a global maximum when θ = C, and a local maximum when θ = A. This demonstrates the importance of good initial guesses for the parameters. Any initial guesses to the left of B are likely to lead to the selection of A rather than C. The situation is likely to be even worse in practice, since the log-likelihood function will be maximised with respect to several parameters, rather than one, and there could be many local optima. Another possibility that would make optimisation difficult is when the LLF is flat around the maximum. So, for example, if the peak corresponding to C in Figure 9.2, were flat rather than sharp, a range of values for θ could lead to very similar values for the LLF, making it difficult to choose between them. So, to explain again in more detail, the optimisation is done in the way shown in Box 9.3. The optimisation methods employed by many software packages such as EViews are based on the determination of the first and second derivatives of the log-likelihood function with respect to the parameter values at each iteration, known as the gradient and Hessian (the matrix of second derivatives of the LLF w.r.t the parameters), respectively. An algorithm for optimisation due to Berndt et al. (1974), known as BHHH, is available in EViews. BHHH employs only first derivatives (calculated numerically rather than analytically) and approximations to the second derivatives are calculated. Not calculating the actual Hessian at each iteration at each time step increases computational speed, but the 518

approximation may be poor when the LLF is a long way from its maximum value, requiring more iterations to reach the optimum. The Marquardt algorithm, available in EViews, is a modification of BHHH (both of which are variants on the Gauss–Newton method) that incorporates a ‘correction’, the effect of which is to push the coefficient estimates more quickly to their optimal values. All of these optimisation methods are described in detail in Press et al. (1992). BOX 9.3 Using maximum likelihood estimation in practice (1) Set up the LLF. (2) Use regression to get initial estimates for the mean parameters. (3) Choose some initial guesses for the conditional variance parameters. In most software packages, the default initial values for the conditional variance parameters would be zero. This is unfortunate since zero parameter values often yield a local maximum of the likelihood function. So if possible, set plausible initial values away from zero. (4) Specify a convergence criterion – either by criterion or by value. When ‘by criterion’ is selected, the package will continue to search for ‘better’ parameter values that give a higher value of the LLF until the change in the value of the LLF between iterations is less than the specified convergence criterion. Choosing ‘by value’ will lead to the software searching until the change in the coefficient estimates are small enough. For example, the default convergence criterion for EViews is 0.001, which means that convergence is achieved and the program will stop searching if the biggest percentage change in any of the coefficient estimates for the most recent iteration is smaller than 0.1%.

9.9.2 Non-Normality and Maximum Likelihood Recall that the conditional normality assumption for ut is essential in specifying the likelihood function. It is possible to test for non-normality using the following representation (9.44)

519

(9.45) Note that one would not expect ut to be normally distributed – it is a disturbance term from the regression model, which will imply it is likely to have fat tails. A plausible method to test for normality would be to construct the statistic (9.46) which would be the model disturbance at each point in time t divided by the conditional standard deviation at that point in time. Thus, it is the vt that are assumed to be normally distributed, not ut. The sample counterpart would be (9.47) which is known as a standardised residual. Whether the are normal can be examined using any standard normality test, such as the Bera–Jarque. Typically, are still found to be leptokurtic, although less so than the The upshot is that the GARCH model is able to capture some, although not all, of the leptokurtosis in the unconditional distribution of asset returns. Is it a problem if are not normally distributed? Well, the answer is ‘not really’. Even if the conditional normality assumption does not hold, the parameter estimates will still be consistent if the equations for the mean and variance are correctly specified. However, in the context of nonnormality, the usual standard error estimates will be inappropriate, and a different variance–covariance matrix estimator that is robust to nonnormality, due to Bollerslev and Wooldridge (1992), should be used. This procedure (i.e., maximum likelihood with Bollerslev–Wooldridge standard errors) is known as quasi-maximum likelihood, or QML.

9.10 Extensions to the Basic GARCH Model Since the GARCH model was developed, a huge number of extensions and variants have been proposed. A couple of the most important examples will be highlighted here. Interested readers who wish to investigate further are directed to a comprehensive survey by Bollerslev, Chou and Kroner (1992). 520

Many of the extensions to the GARCH model have been suggested as a consequence of perceived problems with standard GARCH(p, q) models. First, the non-negativity conditions may be violated by the estimated model. The only way to avoid this for sure would be to place artificial constraints on the model coefficients in order to force them to be nonnegative. Second, GARCH models cannot account for leverage effects (explained below), although they can account for volatility clustering and leptokurtosis in a series. Finally, the model does not allow for any direct feedback between the conditional variance and the conditional mean. Some of the most widely used and influential modifications to the model will now be examined. These may remove some of the restrictions or limitations of the basic model.

9.11 Asymmetric GARCH Models One of the primary restrictions of GARCH models is that they enforce a symmetric response of volatility to positive and negative shocks. This arises since the conditional variance in equations such as equation (9.39) is a function of the magnitudes of the lagged residuals and not their signs (in other words, by squaring the lagged error in equation (9.39), the sign is lost). However, it has been argued that a negative shock to financial time series is likely to cause volatility to rise by more than a positive shock of the same magnitude. In the case of equity returns, such asymmetries are typically attributed to leverage effects, whereby a fall in the value of a firm’s stock causes the firm’s debt to equity ratio to rise. This leads shareholders, who bear the residual risk of the firm, to perceive their future cashflow stream as being relatively more risky. An alternative view is provided by the ‘volatility-feedback’ hypothesis. Assuming constant dividends, if expected returns increase when stock price volatility increases, then stock prices should fall when volatility rises. Although asymmetries in returns series other than equities cannot be attributed to changing leverage, there is equally no reason to suppose that such asymmetries only exist in equity returns. Two popular asymmetric formulations are explained below: the GJR model, named after the authors Glosten, Jagannathan and Runkle (1993), and the exponential GARCH (EGARCH) model proposed by Nelson (1991).

9.12 The GJR model 521

The GJR model is a simple extension of GARCH with an additional term added to account for possible asymmetries. The conditional variance is now given by (9.48) where It−1 = 1 if ut−1 < 0 = 0 otherwise For a leverage effect, we would see γ > 0. Notice now that the condition for nonnegativity will be α0 > 0, α1 > 0, β ≥ 0, and α1 + γ ≥ 0. That is, the model is still admissible, even if γ < 0, provided that α1 + γ ≥ 0. EXAMPLE 9.1 To offer an illustration of the GJR approach, using monthly S&P500 returns from December 1979 until June 1998, the following results would be obtained, with t-ratios in parentheses (9.49) (9.50) Note that the asymmetry term, γ, has the correct sign and is significant. To see how volatility rises more after a large negative shock than a large positive one, suppose that and consider If this implies that However, a shock of the same magnitude but of opposite sign, implies that the fitted conditional variance for time t will be

9.13 The EGARCH Model The exponential GARCH model was proposed by Nelson (1991). There are various ways to express the conditional variance equation, but one possible specification is given by

522

(9.51)

The model has several advantages over the pure GARCH specification. First, since the is modelled, then even if the parameters are negative, will be positive. There is thus no need to artificially impose nonnegativity constraints on the model parameters. Second, asymmetries are allowed for under the EGARCH formulation, since if the relationship between volatility and returns is negative, γ, will be negative. Note that in the original formulation, Nelson assumed a generalised error distribution (GED) structure for the errors. GED is a very broad family of distributions that can be used for many types of series. However, owing to its computational ease and intuitive interpretation, almost all applications of EGARCH employ conditionally normal errors as discussed above rather than using GED.

9.14 Tests for Asymmetries in Volatility Engle and Ng (1993) have proposed a set of tests for asymmetry in volatility, known as sign and size bias tests. The Engle and Ng tests should thus be used to determine whether an asymmetric model is required for a given series, or whether the symmetric GARCH model can be deemed adequate. In practice, the Engle–Ng tests are usually applied to the residuals of a GARCH fit to the returns data. Define as an indicator dummy that takes the value 1 if and zero otherwise. The test for sign bias is based on the significance or otherwise of ϕ1 in (9.52) where υt is an iid error term. If positive and negative shocks to impact differently upon the conditional variance, then ϕ1 will be statistically significant. It could also be the case that the magnitude or size of the shock will affect whether the response of volatility to shocks is symmetric or not. In this case, a negative size bias test would be conducted, based on a regression where is now used as a slope dummy variable. Negative size bias is argued to be present if ϕ1 is statistically significant in the regression 523

(9.53) Finally, defining so that picks out the observations with positive innovations, Engle and Ng propose a joint test for sign and size bias based on the regression (9.54) Significance of ϕ1 indicates the presence of sign bias, where positive and negative shocks have differing impacts upon future volatility, compared with the symmetric response required by the standard GARCH formulation. On the other hand, the significance of ϕ2 or ϕ3 would suggest the presence of size bias, where not only the sign but the magnitude of the shock is important. A joint test statistic is formulated in the standard fashion by calculating TR2 from regression equation (9.54), which will asymptotically follow a χ2 distribution with three degrees of freedom under the null hypothesis of no asymmetric effects.

9.14.1 News Impact Curves A pictorial representation of the degree of asymmetry of volatility to positive and negative shocks is given by the news impact curve introduced by Pagan and Schwert (1990). The news impact curve plots the nextperiod volatility that would arise from various positive and negative values of ut−1, given an estimated model. The curves are drawn by using the estimated conditional variance equation for the model under consideration, with its given coefficient estimates, and with the lagged conditional variance set to the unconditional variance. Then, successive values of ut−1 are used in the equation to determine what the corresponding values of derived from the model would be. For example, suppose that model estimates are constructed for GARCH and GJR models for S&P500 data. Values of ut−1 in the range (−1, +1) are substituted into the equations in each case to investigate the impact on the conditional variance during the next period. The resulting news impact curves for the GARCH and GJR models are given in Figure 9.3.

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Figure 9.3 News impact curves for S&P500 returns using coefficients

implied from GARCH and GJR model estimates As can be seen from Figure 9.3, the GARCH news impact curve (the light blue line) is of course symmetrical about zero, so that a shock of given magnitude will have the same impact on the future conditional variance whatever its sign. On the other hand, the GJR news impact curve (the dark blue line) is asymmetric, with negative shocks having more impact on future volatility than positive shocks of the same magnitude. It can also be seen that a negative shock of given magnitude will have a bigger impact under GJR than would be implied by a GARCH model, while a positive shock of given magnitude will have more impact under GARCH than GJR. The latter result arises as a result of the reduction in the value of α1, the coefficient on the lagged squared error, when the asymmetry term is included in the model.

9.15 GARCH-in-Mean Most models used in finance suppose that investors should be rewarded for taking additional risk by obtaining a higher return. One way to operationalise this concept is to let the return of a security be partly determined by its risk. Engle, Lilien and Robins (1987) suggested an ARCH-M specification, where the conditional variance of asset returns enters into the conditional mean equation. Since GARCH models are now 525

considerably more popular than ARCH, it is more common to estimate a GARCH-M model. An example of a GARCH-M model is given by the specification (9.55) (9.56) If δ is positive and statistically significant, then increased risk, given by an increase in the conditional variance, leads to a rise in the mean return. Thus δ can be interpreted as a risk premium. In some empirical applications, the conditional variance term, appears directly in the conditional mean equation, rather than in square root form, σt−1. Also, in some applications the term is contemporaneous, rather than lagged.

9.16 Uses of GARCH-Type Models Including Volatility Forecasting Essentially GARCH models are useful because they can be used to model the volatility of a series over time. It is possible to combine together more than one of the time series models that have been considered so far in this book, to obtain more complex ‘hybrid’ models. Such models can account for a number of important features of financial series at the same time – e.g., an ARMA–EGARCH(1,1)-M model; the potential complexity of the model is limited only by the imagination! GARCH-type models can be used to forecast volatility. GARCH is a model to describe movements in the conditional variance of an error term, ut, which may not appear particularly useful. But it is possible to show that (9.57) So the conditional variance of y, given its previous values, is the same as the conditional variance of u, given its previous values. Hence, modelling will give models and forecasts for the variance of yt as well. Thus, if the dependent variable in a regression, yt is an asset return series, forecasts of will be forecasts of the future variance of yt. So one primary usage of GARCH-type models is in forecasting volatility. This can be useful in, for example, the pricing of financial options where volatility is an input to the pricing model. For example, the value of a ‘plain vanilla’ call option is a function of the current value of the underlying, the strike price, the time to 526

maturity, the risk-free interest rate and volatility. The required volatility, to obtain an appropriate options price, is really the volatility of the underlying asset expected over the lifetime of the option. As stated previously, it is possible to use a simple historical average measure as the forecast of future volatility, but another method that seems more appropriate would be to use a time series model such as GARCH to compute the volatility forecasts. The forecasting ability of various models is considered in a paper by Day and Lewis (1992), discussed in detail below. Producing forecasts from models of the GARCH class is relatively simple, and the algebra involved is very similar to that required to obtain forecasts from ARMA models. An illustration is given by Example 9.2. EXAMPLE 9.2 Consider the following GARCH(1,1) model (9.58) (9.59) Suppose that the researcher had estimated the above GARCH model for a series of returns on a stock index and obtained the following parameter estimates: If the researcher has data available up to and including time T, write down a set of equations in and and their lagged values,which could be employed to produce one-, two-, and three-step-ahead forecasts for the conditional variance of yt. What is needed is to generate forecasts of σT+12|ΩT, σT+22|ΩT, …, σT+s2|ΩT where ΩT denotes all information available up to and including observation T. For time T, the conditional variance equation is given by equation (9.59). Adding one to each of the time subscripts of this equation, and then two, and then three would yield equations (9.60)–(9.62) (9.60) (9.61) (9.62) Let be the one-step-ahead forecast for σ2 made at time T. This is easy to calculate since, at time T, the values of all the terms on the RHS 527

are known. would be obtained by taking the conditional expectation of equation (9.60). Given how is the two-step-ahead forecast for σ2 made at time T, calculated? (9.63) From equation (9.61), it is possible to write (9.64) where is the expectation, made at time T, of squared disturbance term. It is necessary to find using the expression for the variance of a random variable ut. The model assumes that the series ut has zero mean, so that the variance can be written (9.65) The conditional variance of ut is

so (9.66)

Turning this argument around, and applying it to the problem at hand (9.67) but

is not known at time T, so it is replaced with the forecast for it, so that equation (9.64) becomes (9.68) (9.69)

What about the three-step-ahead forecast? By similar arguments, (9.70) (9.71) (9.72) 528

(9.73) Any s-step-ahead forecasts would be produced by (9.74) for any value of s ≥ 2. It is worth noting at this point that variances, and therefore variance forecasts, are additive over time. This is a very useful property. Suppose, for example, that using daily foreign exchange returns, one-, two-, three-, four-, and five-step-ahead variance forecasts have been produced, i.e., a forecast has been constructed for each day of the next trading week. The forecasted variance for the whole week would simply be the sum of the five daily variance forecasts. If the standard deviation is the required volatility estimate rather than the variance, simply take the square root of the variance forecasts. Note also, however, that standard deviations are not additive. Hence, if daily standard deviations are the required volatility measure, they must be squared to turn them to variances. Then the variances would be added and the square root taken to obtain a weekly standard deviation.

9.17 Testing Non-Linear Restrictions or Testing Hypotheses About Non-Linear Models The usual t- and F-tests are still valid in the context of non-linear models, but they are not flexible enough. For example, suppose that it is of interest to test a hypothesis that α1β = 1. Now that the model class has been extended to non-linear models, there is no reason to suppose that relevant restrictions are only linear. Under OLS estimation, the F-test procedure works by examining the degree to which the RSS rises when the restrictions are imposed. In very general terms, hypothesis testing under ML works in a similar fashion – that is, the procedure works by examining the degree to which the maximal value of the LLF falls upon imposing the restriction. If the LLF falls ‘a lot’, it would be concluded that the restrictions are not supported by the data and thus the hypothesis should be rejected. There are three hypothesis testing procedures based on maximum 529

likelihood principles: Wald, Likelihood ratio and Lagrange Multiplier. To illustrate briefly how each of these operates, consider a single parameter, θ to be estimated, and denote the ML estimate as and a restricted estimate as Denoting the maximised value of the LLF by unconstrained ML as and the constrained optimum as the three testing procedures can be illustrated as in Figure 9.4.

Figure 9.4 Three approaches to hypothesis testing under maximum

likelihood The tests all require the measurement of the ‘distance’ between the points A (representing the unconstrained maximised value of the log likelihood function) and B (representing the constrained value). The vertical distance forms the basis of the LR test. Twice this vertical distance is given by where L denotes the log-likelihood function, and l denotes the likelihood function. The Wald test is based on the horizontal distance between and while the LM test compares the slopes of the curve at A and B. At A, the unrestricted maximum of the loglikelihood function, the slope of the curve is zero. But is it ‘significantly steep’ at i.e., at point B ? The steeper the curve is at B, the less likely the restriction is to be supported by the data. Expressions for LM test statistics involve the first and second derivatives of the log-likelihood function with respect to the parameters at the constrained estimate. The first derivatives of the log-likelihood function are collectively known as the score vector, measuring the slope of the LLF for each possible value of the parameters. The expected values of 530

the second derivatives comprise the information matrix, measuring the peakedness of the LLF, and how much higher the LLF value is at the optimum than in other places. This matrix of second derivatives is also used to construct the coefficient standard errors. The LM test involves estimating only a restricted regression, since the slope of the LLF at the maximum will be zero by definition. Since the restricted regression is usually easier to estimate than the unrestricted case, LM tests are usually the easiest of the three procedures to employ in practice. The reason that restricted regressions are usually simpler is that imposing the restrictions often means that some components in the model will be set to zero or combined under the null hypothesis, so that there are fewer parameters to estimate. The Wald test involves estimating only an unrestricted regression, and the usual OLS t-tests and F-tests are examples of Wald tests (since again, only unrestricted estimation occurs). Of the three approaches to hypothesis testing in the maximumlikelihood framework, the likelihood ratio test is the most intuitively appealing, and therefore a deeper examination of it will be the subject of the following section; see Ghosh (1991, Section 10.3) for further details.

9.17.1 Likelihood Ratio Tests Likelihood ratio (LR) tests involve estimation under the null hypothesis and under the alternative, so that two models are estimated: an unrestricted model and a model where the restrictions have been imposed. The maximised values of the LLF for the restricted and unrestricted cases are ‘compared’. Suppose that the unconstrained model has been estimated and that a given maximised value of the LLF, denoted Lu, has been achieved. Suppose also that the model has been estimated imposing the constraint(s) and a new value of the LLF obtained, denoted Lr. The LR test statistic asymptotically follows a Chi-squared distribution and is given by (9.75) where m = number of restrictions. Note that the maximised value of the log-likelihood function will always be at least as big for the unrestricted model as for the restricted model, so that Lr ≤ Lu. This rule is intuitive and comparable to the effect of imposing a restriction on a linear model estimated by OLS, that RRSS ≥ URSS. Similarly, the equality between Lr and Lu will hold only when the restriction was already present in the data. 531

Note, however, that the usual F-test is in fact a Wald test, and not a LR test – that is, it can be calculated using an unrestricted model only. The F-test approach based on comparing RSS arises conveniently as a result of the OLS algebra. Example 9.3 demonstrates how a likelihood ratio test is implemented. EXAMPLE 9.3 A GARCH model is estimated and a maximised LLF of 66.85 is obtained. Suppose that a researcher wishes to test whether β = 0 in (9.77) (9.76) (9.77) The model is estimated imposing the restriction and the maximised LLF falls to 64.54. Is the restriction supported by the data, which would correspond to the situation where an ARCH(1) specification was sufficient? The test statistic is given by (9.78) The test follows a χ2(1) = 3.84 at 5%, so that the null is marginally rejected. It would thus be concluded that an ARCH(1) model, with no lag of the conditional variance in the variance equation, is not quite sufficient to describe the dependence in volatility over time.

9.18 Volatility Forecasting: Some Examples and Results from the Literature There is a vast and relatively new literature that attempts to compare the accuracies of various models for producing out-of-sample volatility forecasts. Akgiray (1989), for example, finds the GARCH model superior to ARCH, exponentially weighted moving average and historical mean models for forecasting monthly US stock index volatility. A similar result concerning the apparent superiority of GARCH is observed by West and Cho (1995) using one-step-ahead forecasts of dollar exchange rate volatility, although for longer horizons, the model behaves no better than their alternatives. Pagan and Schwert (1990) compare GARCH, EGARCH, 532

Markov switching regime and three non-parametric models for forecasting monthly US stock return volatilities. The EGARCH followed by the GARCH models perform moderately; the remaining models produce very poor predictions. Franses and van Dijk (1996) compare three members of the GARCH family (standard GARCH, QGARCH and the GJR model) for forecasting the weekly volatility of various European stock market indices. They find that the non-linear GARCH models were unable to beat the standard GARCH model. Finally, Brailsford and Faff (1996) find GJR and GARCH models slightly superior to various simpler models for predicting Australian monthly stock index volatility. The conclusion arising from this growing body of research is that forecasting volatility is a ‘notoriously difficult task’ (Brailsford and Faff, 1996, p. 419), although it appears that conditional heteroscedasticity models are among the best that are currently available. In particular, more complex non-linear and non-parametric models are inferior in prediction to simpler models, a result echoed in an earlier paper by Dimson and Marsh (1990) in the context of relatively complex versus parsimonious linear models. Finally, Brooks (1998), considers whether measures of market volume can assist in improving volatility forecast accuracy, finding that they cannot. A particularly clear example of the style and content of this class of research is given by Day and Lewis (1992). The Day and Lewis study will therefore now be examined in depth. The purpose of their paper is to consider the out-of-sample forecasting performance of GARCH and EGARCH models for predicting stock index volatility. The forecasts from these econometric models are compared with those given from an ‘implied volatility’. As discussed above, implied volatility is the market’s expectation of the ‘average’ level of volatility of an underlying asset over the life of the option that is implied by the current traded price of the option. Given an assumed model for pricing options, such as the Black– Scholes, all of the inputs to the model except for volatility can be observed directly from the market or are specified in the terms of the option contract. Thus, it is possible, using an iterative search procedure such as the Newton–Raphson method (see, for example, Watsham and Parramore, 2004), to ‘back out’ the volatility of the underlying asset from the option’s price. An important question for research is whether implied or econometric models produce more accurate forecasts of the volatility of the underlying asset. If the options and underlying asset markets are informationally efficient, econometric volatility forecasting models based on past realised values of underlying volatility should have no incremental explanatory power for future values of volatility of the underlying asset. 533

On the other hand, if econometric models do hold additional information useful for forecasting future volatility, it is possible that such forecasts could be turned into a profitable trading rule. The data employed by Day and Lewis comprise weekly closing prices (Wednesday to Wednesday, and Friday to Friday) for the S&P100 Index option and the underlying index from 11 March 1983–31 December 1989. They employ both mid-week to midweek returns and Friday to Friday returns to determine whether weekend effects have any significant impact on the latter. They argue that Friday returns contain expiration effects since implied volatilities are seen to jump on the Friday of the week of expiration. This issue is not of direct interest to this book, and consequently only the mid-week to mid-week results will be shown here. The models that Day and Lewis employ are as follows. First, for the conditional mean of the time series models, they employ a GARCH-M specification for the excess of the market return over a risk-free proxy (9.79) where RMt denotes the return on the market portfolio, and RFt denotes the riskfree rate. Note that Day and Lewis denote the conditional variance by while this is modified to the standard ht here. Also, the notation will be used to denote implied volatility estimates. For the variance, two specifications are employed: a ‘plain vanilla’ GARCH(1,1) and an EGARCH (9.80) or (9.81) One way to test whether implied or GARCH-type volatility models perform best is to add a lagged value of the implied volatility estimate to equations (9.80) and (9.81). A ‘hybrid’ or ‘encompassing’ specification would thus result. Equation (9.80) becomes (9.82) and equation (9.81) becomes 534

(9.83)

The tests of interest are given by H0 : δ = 0 in (9.82) or (9.83). If these null hypotheses cannot be rejected, the conclusion would be that implied volatility contains no incremental information useful for explaining volatility than that derived from a GARCH model. At the same time, H0 : α1 = 0 and β1 = 0 in (9.82), and H0: α1 = 0 and β1 = 0 and θ = 0 and γ = 0 in (9.83) are also tested. If this second set of restrictions holds, then equations (9.82) and (9.83) collapse to (9.82′) and (9.83′) These sets of restrictions on equations (9.82) and (9.83) test whether the lagged squared error and lagged conditional variance from a GARCH model contain any additional explanatory power once implied volatility is included in the specification. All of these restrictions can be tested fairly easily using a likelihood ratio test. The results of such a test are presented in Table 9.1. Table 9.1 GARCH versus implied volatility

535

Notes: t-ratios in parentheses, Log-L denotes the maximised value of the log-likelihood function in each case. χ2 denotes the value of the test statistic, which follows a χ2(1) in the case of equation (9.82) restricted to equation (9.80), and a χ2(2) in the case of equation (9.82) restricted to equation (9.82′). Source: Day and Lewis (1992). Reprinted with the permission of Elsevier.

It appears from the coefficient estimates and their standard errors under the specification (9.82) that the implied volatility term (δ) is statistically significant, while the GARCH terms (α1 and β1) are not. However, the test statistics given in the final column are both greater than their corresponding χ2 critical values, indicating that both GARCH and implied volatility have incremental power for modelling the underlying stock volatility. A similar analysis is undertaken in Day and Lewis that compares EGARCH with implied volatility. The results are presented here in Table 9.2. Table 9.2 EGARCH versus implied volatility

536

Notes: t-ratios in parentheses, Log-L denotes the maximised value of the log-likelihood function in each case. χ2 denotes the value of the test statistic, which follows a χ2(1) in the case of equation (9.83) restricted to equation (9.81), and a χ2(3) in the case of equation (9.83) restricted to equation (9.83′). Source: Day and Lewis (1992). Reprinted with the permission of Elsevier.

The EGARCH results tell a very similar story to those of the GARCH specifications. Neither the lagged information from the EGARCH specification nor the lagged implied volatility terms can be suppressed, according to the likelihood ratio statistics. In specification (9.83), both the EGARCH terms and the implied volatility coefficients are marginally significant. However, the tests given above do not represent a true test of the predictive ability of the models, since all of the observations were used in both estimating and testing the models. Hence the authors proceed to conduct an out-of-sample forecasting test. There are a total of 729 data points in their sample. They use the first 410 to estimate the models, and then make a one-step-ahead forecast of the following week’s volatility. They then roll the sample forward one observation at a time, constructing a new one-step-ahead forecast at each stage. They evaluate the forecasts in two ways. The first is by regressing the realised volatility series on the forecasts plus a constant (9.84) 537

where is the ‘actual’ value of volatility at time t + 1, and is the value forecasted for it during period t. Perfectly efficient forecasts would imply b0 = 0 and b1 = 1. The second method is via a set of forecast encompassing tests. Essentially, these operate by regressing the realised volatility on the forecasts generated by several models. The forecast series that have significant coefficients are concluded to encompass those of models whose coefficients are not significant. But what is volatility? In other words, with what measure of realised or ‘ex post’ volatility should the forecasts be compared? This is a question that received very little attention in the literature until recently. A common method employed is to assume, for a daily volatility forecasting exercise, that the relevant ex post measure is the square of that day’s return. For any random variable rt, its conditional variance can be expressed as (9.85) As stated previously, it is typical, and not unreasonable for relatively high frequency data, to assume that E(rt) is zero, so that the expression for the variance reduces to (9.86) Andersen and Bollerslev (1998) argue that squared daily returns provide a very noisy proxy for the true volatility, and a much better proxy for the day’s variance would be to compute the volatility for the day from intradaily data. For example, a superior daily variance measure could be obtained by taking hourly returns, squaring them and adding them up. The reason that the use of higher frequency data provides a better measure of ex post volatility is simply that it employs more information. By using only daily data to compute a daily volatility measure, effectively only two observations on the underlying price series are employed. If the daily closing price is the same one day as the next, the squared return and therefore the volatility would be calculated to be zero, when there may have been substantial intra-day fluctuations. Hansen and Lunde (2006) go further and suggest that even the ranking of models by volatility forecast accuracy could be inconsistent if the evaluation uses a poor proxy for the true, underlying volatility. Day and Lewis use two measures of ex post volatility in their study (for which the frequency of data employed in the models is weekly) 538

(1) The square of the weekly return on the index, which they call SR (2) The variance of the week’s daily returns multiplied by the number of trading days in that week, which they call WV. The Andersen and Bollerslev argument implies that the latter measure is likely to be superior, and therefore that more emphasis should be placed on those results. The results for the separate regressions of realised volatility on a constant and the forecast are given in Table 9.3. The coefficient estimates for b0 given in Table 9.3 can be interpreted as indicators of whether the respective forecasting approaches are biased. In all cases, the b0 coefficients are close to zero. Only for the historic volatility forecasts and the implied volatility forecast when the ex post measure is the squared weekly return, are the estimates statistically significant. Positive coefficient estimates would suggest that on average the forecasts are too low. The estimated b1 coefficients are in all cases a long way from unity, except for the GARCH (with daily variance ex post volatility) and EGARCH (with squared weekly variance as ex post measure) models. Finally, the R2 values are very small (all less than 10%, and most less than 3%), suggesting that the forecast series do a poor job of explaining the variability of the realised volatility measure. Table 9.3 Out-of-sample predictive power for weekly volatility forecasts

539

Notes: ‘Historic’ refers to the use of a simple historical average of the squared returns to forecast volatility; t-ratios in parentheses; SR and WV refer to the square of the weekly return on the S&P100, and the variance of the week’s daily returns multiplied by the number of trading days in that week, respectively. Source: Day and Lewis (1992). Reprinted with the permission of Elsevier.

The forecast encompassing regressions are based on a procedure due to Fair and Shiller (1990) that seeks to determine whether differing sets of forecasts contain different sets of information from one another. The test regression is of the form (9.87) with results presented in Table 9.4. Table 9.4 Comparisons of the relative information content of out-ofsample volatility forecasts

540

Notes: t-ratios in parentheses; the ex post measure used in this table is the variance of the week’s daily returns multiplied by the number of trading days in that week. Source: Day and Lewis (1992). Reprinted with the permission of Elsevier.

The sizes and significances of the coefficients in Table 9.4 are of interest. The most salient feature is the lack of significance of most of the forecast series. In the first comparison, neither the implied nor the GARCH forecast series have statistically significant coefficients. When historical volatility is added, its coefficient is positive and statistically significant. An identical pattern emerges when forecasts from implied and EGARCH models are compared: that is, neither forecast series is significant, but when a simple historical average series is added, its coefficient is significant. It is clear from this, and from the last row of Table 9.4, that the asymmetry term in the EGARCH model has no additional explanatory power compared with that embodied in the symmetric GARCH model. Again, all of the R2 values are very low (less than 4%). The conclusion reached from this study (which is broadly in line with many others) is that within sample, the results suggest that implied volatility contains extra information not contained in the GARCH/EGARCH specifications. But the out-of-sample results suggest that predicting volatility is a difficult task!

9.19 Stochastic Volatility Models Revisited 541

Autoregressive models were discussed above in Section 9.6 and these are special cases of a more general class of models known as stochastic volatility (SV) models. It is a common misconception that GARCH-type specifications are sorts of stochastic volatility models. However, as the name suggests, stochastic volatility models differ from GARCH principally in that the conditional variance equation of a GARCH specification is completely deterministic given all information available up to that of the previous period. In other words, there is no error term in the variance equation of a GARCH model, only in the mean equation. Stochastic volatility models contain a second error term, which enters into the conditional variance equation. The autoregressive volatility specification is simple to understand and simple to estimate, because it requires that we have an observable measure of volatility which is then simply used as any other variable in an autoregressive model. However, the term ‘stochastic volatility’ is usually associated with a different formulation, a possible example of which would be (9.88) (9.89) where ηt is another N(0,1) random variable that is independent of ut. Here the volatility is latent rather than observed, and so is modelled indirectly. Stochastic volatility models are closely related to the financial theories used in the options pricing literature. Early work by Black and Scholes (1973) had assumed that volatility is constant through time. Such an assumption was made largely for simplicity, although it could hardly be considered realistic. One unappealing side-effect of employing a model with the embedded assumption that volatility is fixed, is that options deep in-the-money and far out-of-the-money are underpriced relative to actual traded prices. This empirical observation provided part of the genesis for stochastic volatility models, where the logarithm of an unobserved variance process is modelled by a linear stochastic specification, such as an autoregressive model. The primary advantage of stochastic volatility models is that they can be viewed as discrete time approximations to the continuous time models employed in options pricing frameworks (see, for example, Hull and White, 1987). However, such models are hard to estimate. For reviews of (univariate) stochastic volatility models, see Taylor (1994), Ghysels, Harvey and Renault (1995) or Shephard (1996) and the references therein. 542

While stochastic volatility models have been widely employed in the mathematical options pricing literature, they have not been popular in empirical discrete-time financial applications, probably owing to the complexity involved in the process of estimating the model parameters (see Harvey, Ruiz and Shephard, 1994). So, while GARCH-type models are further from their continuous time theoretical underpinnings than stochastic volatility, they are much simpler to estimate using maximum likelihood. A relatively simple modification to the maximum likelihood procedure used for GARCH model estimation is not available, and hence stochastic volatility models are not discussed further here.

9.19.1 Higher Moment Models Research over the past two decades has moved from examination purely of the first moment of financial time series (i.e., estimating models for the returns themselves), to consideration of the second moment (models for the variance). While this clearly represents a large step forward in the analysis of financial data, it is also evident that conditional variance specifications are not able to fully capture all of the relevant time series properties. For example, GARCH models with normal (0,1) standardised disturbances cannot generate sufficiently fat tails to model the leptokurtosis that is actually observed in financial asset returns series. One proposed approach to this issue has been to suggest that the standardised disturbances are drawn from a Student’s t distribution rather than a normal. However, there is also no reason to suppose that the fatness of tails should be constant over time, which it is forced to be by the GARCH-t model. Another possible extension would be to use a conditional model for the third or fourth moments of the distribution of returns (i.e., the skewness and kurtosis, respectively). Under such a specification, the conditional skewness or kurtosis of the returns could follow a GARCH-type process that allows it to vary through time. Harvey and Siddique (1999, 2000) have developed an autoregressive conditional skewness model, while a conditional kurtosis model was proposed in Brooks et al. (2005). Such models could have many other applications in finance, including asset allocation (portfolio selection), option pricing, estimation of risk premia, and so on. An extension of the analysis to moments of the return distribution higher than the second has also been undertaken in the context of the capital asset pricing model, where the conditional co-skewness and cokurtosis of the asset’s returns with the market’s are accounted for (e.g., 543

Hung, Shackleton and Xu, 2004). A recent study by Brooks, Černý and Miffre (2012) proposed a utility-based framework for the determination of optimal hedge ratios that can allow for the impact of higher moments on the hedging decision in the context of hedging commodity exposures with futures contracts.

9.19.2 Tail Models It is widely known that financial asset returns do not follow a normal distribution, but rather they are almost always leptokurtic, or fat-tailed. This observation has several implications for econometric modelling. First, models and inference procedures are required that are robust to nonnormal error distributions. Second, the riskiness of holding a particular security is probably no longer appropriately measured by its variance alone. In a risk management context, assuming normality when returns are fat-tailed will result in a systematic underestimation of the riskiness of the portfolio. Consequently, several approaches have been employed to systematically allow for the leptokurtosis in financial data, including the use of a Student’s t distribution. Arguably the simplest approach is the use of a mixture of normal distributions. It can be seen that a mixture of normal distributions with different variances will lead to an overall series that is leptokurtic. Second, a Student’s t distribution can be used, with the usual degrees of freedom parameter estimated using maximum likelihood along with other parameters of the model. The degrees of freedom estimate will control the fatness of the tails fitted from the model. Other probability distributions can also be employed, such as the ‘stable’ distributions that fall under the general umbrella of extreme value theory – see Section 14.3 of Chapter 14 for a detailed presentation of this class of models.

9.20 Forecasting Covariances and Correlations A major limitation of the volatility models examined above is that they are entirely univariate in nature – that is, they model the conditional variance of each series entirely independently of all other series. This is potentially an important limitation for two reasons. First, to the extent that there may be ‘volatility spillovers’ between markets or assets (a tendency for volatility to change in one market or asset following a change in the volatility of another), the univariate model will be misspecified. For instance, using a multivariate model will allow us to determine whether the 544

volatility in one market leads or lags the volatility in others. Second, it is often the case in finance that the covariances between series are of interest, as well as the variances of the individual series themselves. The calculation of hedge ratios, portfolio value at risk estimates, CAPM betas, and so on, all require covariances as inputs. Multivariate GARCH models can potentially overcome both of these deficiencies with their univariate counterparts. Multivariate extensions to GARCH models can be used to forecast the volatilities of the component series, just as with univariate models and since the volatilities of financial time series often move together, a joint approach to modelling may be more efficient than treating each separately. In addition, because multivariate models give estimates for the conditional covariances as well as the conditional variances, they have a number of other potentially useful applications. Several papers have investigated the forecasting ability of various models incor-porating correlations. Siegel (1997), for example, finds that implied correlation forecasts from traded options encompass all information embodied in the historical returns (although he does not consider EWMA- or GARCH-based models). Walter and Lopez (2000), on the other hand, find that implied correlation is generally less useful for predicting the future correlation between the underlying assets’ returns than forecasts derived from GARCH models. Finally, Gibson and Boyer (1998) find that a diagonal GARCH and a Markov switching approach provide better correlation forecasts than simpler models in the sense that the latter produce smaller profits when the forecasts are employed in a trading strategy.

9.21 Covariance Modelling and Forecasting in Finance: Some Examples 9.21.1 The Estimation of Conditional Betas The CAPM beta for asset i is defined as the ratio of the covariance between the market portfolio return and the asset return, to the variance of the market portfolio return. Betas are typically constructed using a set of historical data on market variances and covariances. However, like most other problems in finance, beta estimation conducted in this fashion is backward-looking, when investors should really be concerned with the beta that will prevail in the future over the time that the investor is considering holding the asset. Multivariate GARCH models provide a 545

simple method for estimating conditional (or time-varying) betas. Then forecasts of the covariance between the asset and the market portfolio returns and forecasts of the variance of the market portfolio are made from the model, so that the beta is a forecast, whose value will vary over time (9.90) where βi,t is the time-varying beta estimate at time t for stock i, σim,t is the covariance between market returns and returns to stock i at time t and is the variance of the market return at time t.

9.21.2 Dynamic Hedge Ratios Although there are many techniques available for reducing and managing risk, the simplest and perhaps the most widely used, is hedging with futures contracts. A hedge is achieved by taking opposite positions in spot and futures markets simultaneously, so that any loss sustained from an adverse price movement in one market should to some degree be offset by a favourable price movement in the other. The ratio of the number of units of the futures asset that are purchased relative to the number of units of the spot asset is known as the hedge ratio. Since risk in this context is usually measured as the volatility of portfolio returns, an intuitively plausible strategy might be to choose that hedge ratio which minimises the variance of the returns of a portfolio containing the spot and futures position; this is known as the optimal hedge ratio. The optimal value of the hedge ratio may be determined in the usual way, following Hull (2017) by first defining: ΔS = change in spot price S, during the life of the hedge ΔF = change in futures price, F, during the life of the hedge σs = standard deviation of ΔSσF = standard deviation of ΔFp = correlation coefficient between ΔS and ΔFh = hedge ratio For a short hedge (i.e., long in the asset and short in the futures contract), the change in the value of the hedger’s position during the life of the hedge will be given by (ΔS − h ΔF), while for a long hedge, the appropriate expression will be (hΔF − ΔS). The variances of the two hedged portfolios (long spot and short futures or long futures and short spot) are the same. These can be obtained from 546

Remembering the rules for manipulating the variance operator, this can be written or

Hence the variance of the change in the value of the hedged position is given by (9.91) Minimising this expression w.r.t. h would give (9.92) Again, according to this formula, the optimal hedge ratio is time-invariant, and would be calculated using historical data. However, what if the standard deviations are changing over time? The standard deviations and the correlation between movements in the spot and futures series could be forecast from a multivariate GARCH model, so that the expression above is replaced by (9.93) Various models are available for covariance or correlation forecasting, and several will be discussed below, which are grouped into simple models, multivariate GARCH models, and specific correlation models.

9.22 Simple Covariance Models 9.22.1 Historical Covariance and Correlation In exactly the same fashion as for volatility, the historical covariance or correlation between two series can be calculated in the standard way using a set of historical data.

9.22.2 Implied Covariance Models 547

Implied covariances can be calculated using options whose payoffs are dependent on more than one underlying asset. The relatively small number of such options that exist limits the circumstances in which implied covariances can be calculated. Examples include rainbow options, ‘crackspread’ options for different grades of oil, and currency options. In the latter case, the implied variance of the cross-currency returns xy is given by (9.94) where and are the implied variances of the x and y returns, respectively, and is the implied covariance between x and y. By substituting the observed option implied volatilities of the three currencies into equation (9.94), the implied covariance is obtained via (9.95) So, for instance, if the implied covariance between USD/DEM and USD/JPY is of interest, then the implied variances of the returns of USD/DEM and USD/JPY, as well as the returns of the cross-currency DEM/JPY, are required so as to obtain the implied covariance using equation (9.94).

9.22.3 Exponentially Weighted Moving Average Model for Covariances Again, as for the case of single series volatility modelling, a EWMA specification is available that gives more weight in the calculation of covariance to recent observations than the estimate based on the simple average. The EWMA model estimates for variances and covariances at time t in the bivariate setup with two returns series x and y may be written as (9.96) where i ≠ j for the covariances and i = j; x = y for the variance specifications. As for the univariate case, the fitted values for h also become the forecasts for subsequent periods. λ(0 < λ < 1) again denotes the decay factor determining the relative weights attached to recent versus less recent observations. this parameter could be estimated (for example, by 548

maximum likelihood), but is often set arbitrarily (– for example, Riskmetrics use a decay factor of 0.97 for monthly data but 0.94 when the data are of daily frequency). This equation can be rewritten as an infinite order function of only the returns by successively substituting out the covariances (9.97) While the EWMA model is probably the simplest way to allow for timevarying variances and covariances, the model is a restricted version of an integrated GARCH (IGARCH) specification, and it does not guarantee the fitted variance-covariance matrix to be positive definite. As a result of the parallel with IGARCH, EWMA models also cannot allow for the observed mean reversion in the volatilities or covariances of asset returns that is particularly prevalent at lower frequencies of observation.

9.23 Multivariate GARCH Models Multivariate GARCH models are in spirit very similar to their univariate counterparts, except that the former also specify equations for how the covariances move over time and are therefore by their nature inherently more complex to specify and estimate. Several different multivariate GARCH formulations have been proposed in the literature, the most popular of which are the VECH, the diagonal VECH and the BEKK models. Each of these and several others is discussed in turn below; for a more detailed discussion, see Kroner and Ng (1998). In each case, there are N assets, whose return variances and covariances are to be modelled.

9.23.1 The VECH model As with univariate GARCH models, the conditional mean equation may be parameterised in any way desired, although it is worth noting that, since the conditional variances are measured about the mean, misspecification of the latter is likely to imply misspecification of the former. To introduce some notation, suppose, that yt (y1t y2t … yNt), is an N × 1 vector of timeseries observations, C is an N(N + 1)/2 column vector of conditional variance and covariance intercepts, and A and B are square parameter matrices of order N(N + 1)/2. A common specification of the VECH model, initially due to Bollerslev, Engle and Wooldridge (1988), is 549

(9.98) where Ht is a N × N conditional variance–covariance matrix, Ξt is a N × 1 innovation (disturbance) vector, ψt−1 represents the information set at time t − 1, and VECH (·) denotes the column-stacking operator applied to the upper portion of the symmetric matrix. In the bivariate case (i.e., N = 2), C will be a 3 × 1 parameter vector, and A and B will be 3 × 3 parameter matrices. The unconditional variance matrix for the VECH will be given by C[I − A − B]−1, where I is an identity matrix of order N(N + 1)/2. Stationarity of the VECH model requires that the eigenvalues of [A + B] are all less than one in absolute value. In order to gain a better understanding of how the VECH model works, the elements for N = 2 are written out below. Define

The VECH operator takes the ‘upper triangular’ portion of a matrix, and stacks each element into a vector with a single column. For example, in the case of VECH(Ht), this becomes

where hiit represent the conditional variances at time t of the two-asset return series (i = 1, 2) used in the model, and hijt (i ≠ j) represent the conditional covariances between the asset returns. In the case of this can be expressed as

550

The VECH model in full is given by (9.99) (9.100) (9.101) Thus, it is clear that the conditional variances and conditional covariances depend on the lagged values of all of the conditional variances of, and conditional covariances between, all of the asset returns in the series, as well as the lagged squared errors and the error cross-products. This unrestricted model is highly parameterised, and it is challenging to estimate. For N = 2 there are 21 parameters (C has 3 elements, A and B each have 9 elements), while for N = 3 there are 78, and N = 4 implies 210 parameters!

9.23.2 The Diagonal VECH Model As the number of assets employed increases, estimation of the VECH model can quickly become infeasible. Hence the VECH model’s conditional variance–covariance matrix has been restricted to the form developed by Bollerslev, Engle and Wooldridge (1988), in which A and B are assumed to be diagonal. This restriction implies that there are no direct volatility spillovers from one series to another, which considerably reduces the number of parameters to be estimated to nine in the bivariate case (now A and B each have three elements) and 18 for a trivariate system (i.e., if N = 3). The model, known as a diagonal VECH, is now characterised by

551

(9.102) where ωij, αij and βij are parameters. The diagonal VECH multivariate GARCH model could also be expressed as an infinite order multivariate ARCH model, where the covariance is expressed as a geometrically declining weighted average of past cross products of unexpected returns, with recent observations carrying higher weights. An alternative solution to the dimensionality problem would be to use orthogonal GARCH (see, for example, Van der Weide, 2002) or factor GARCH models (see Engle, Ng and Rothschild, 1990). A disadvantage of the VECH model is that there is no guarantee of a positive semi-definite covariance matrix. A variance–covariance or correlation matrix must always be ‘positive semi-definite’, and in the case where all the returns in a particular series are all the same so that their variance is zero is disregarded, then the matrix will be positive definite. Among other things, this means that the variance–covariance matrix will have all positive numbers on the leading diagonal, and will be symmetrical about this leading diagonal. These properties are intuitively appealing as well as important from a mathematical point of view, for variances can never be negative, and the covariance between two series is the same irrespective of which of the two series is taken first, and positive definiteness ensures that this is the case. A positive definite correlations matrix is also important for many applications in finance – for example, from a risk management point of view. It is this property which ensures that, whatever the weight of each series in the asset portfolio, an estimated value-at-risk is always positive. Fortunately, this desirable property is automatically a feature of timeinvariant correlations matrices which are computed directly using actual data. An anomaly arises when either the correlation matrix is estimated using a non-linear optimisation procedure (as multivariate GARCH models are), or when modified values for some of the correlations are used by the risk manager. The resulting modified correlation matrix may or may not be positive definite, depending on the values of the correlations that are put in, and the values of the remaining correlations. If, by chance, the matrix is not positive definite, the upshot is that for some weightings of the individual assets in the portfolio, the estimated portfolio variance could be negative.

9.23.3 The BEKK model 552

The BEKK model (Engle and Kroner, 1995) addresses the difficulty with VECH of ensuring that the H matrix is always positive definite.1 It is represented by (9.103) where A, and B are N × N matrices of parameters and W is an upper triangular matrix of parameters. The positive definiteness of the covariance matrix is ensured owing to the quadratic nature of the terms on the equation’s RHS.

9.23.4 Model Estimation for Multivariate GARCH Under the assumption of conditional normality, the parameters of the multivariate GARCH models of any of the above specifications can be estimated by maximising the log-likelihood function (9.104) where θ denotes all the unknown parameters to be estimated, N is the number of assets (i.e., the number of series in the system) and T is the number of observations and all other notation is as above. The maximumlikelihood estimate for θ is asymptotically normal, and thus traditional procedures for statistical inference are applicable. Further details on maximum-likelihood estimation in the context of multivariate GARCH models are beyond the scope of this book. But suffice to say that the additional complexity and extra parameters involved compared with univariate models make estimation a computationally more difficult task, although the principles are essentially the same.

9.24 Direct Correlation Models The VECH and BEKK models specify the dynamics of the covariances between a set of series, and the correlations between any given pair of series at each point in time can be constructed by dividing the conditional covariances by the product of the conditional standard deviations. A subtly different approach would be to model the dynamics for the correlations directly – Bauwens, Laurent and Rombouts (2006) term these ‘non-linear combinations of univariate GARCH models’ for reasons that will become 553

apparent in the following sub-section.

9.24.1 The Constant Correlation Model An alternative method for reducing the number of parameters in the MGARCH framework is to require the correlations between the disturbances, ϵt, (or equivalently between the observed variables, yt) to be fixed through time. Thus, although the conditional covariances are not fixed, they are tied to the variances as proposed in the constant conditional correlation (CCC) model due to Bollerslev (1990). The conditional variances in the fixed correlation model are identical to those of a set of univariate GARCH specifications (although they are estimated jointly) (9.105) The off-diagonal elements of Ht, hij,t (i ≠ j), are defined indirectly via the correlations, denoted ρij (9.106) Is it empirically plausible to assume that the correlations are constant through time? Several tests of this assumption have been developed, including a test based on the information matrix due to Bera and Kim (2002) and a Lagrange Multiplier test due to Tse (2000). The conclusions reached appear dependent on which test is used, but there seems to be nonnegligible evidence against constant correlations, particularly in the context of stock returns.

9.24.2 The Dynamic Conditional Correlation Model Several different formulations of the dynamic conditional correlation (DCC) model are available, but a popular specification is due to Engle (2002). The model is related to the CCC formulation described above, but where the correlations are allowed to vary over time. Define the variance– covariance matrix, Ht, as (9.107) where Dt is a diagonal matrix containing the conditional standard deviations (i.e., the square roots of the conditional variances from 554

univariate GARCH model estimations on each of the N individual series) on the leading diagonal; Rt is the conditional correlation matrix. Forcing Rt to be time-invariant would lead back to the constant conditional correlation model. Numerous explicit parameterisations of Rt are possible, including an exponential smoothing approach discussed in Engle (2002). More generally, a model of the MGARCH form could be specified as (9.108) where S is the unconditional correlation matrix of the vector of standardised residuals (from the first stage estimation – see below), ι is a vector of ones, and Qt is an N × N symmetric positive definite variance-covariance matrix. ◦ denotes the Hadamard or elementby-element matrix multiplication procedure. This specification for the intercept term simplifies estimation and reduces the number of parameters to be estimated, but is not necessary. Engle (2002) proposes a GARCHesque formulation for dynamically modelling Qt with the conditional correlation matrix, Rt, then constructed as (9.109) where diag(·) denotes a matrix comprising the main diagonal elements of (·) and Q* is a matrix that takes the square roots of each element in Q. This operation is effectively taking the covariances in Qt and dividing them by the product of the appropriate standard deviations in to create a matrix of correlations. A slightly different form of the DCC was proposed by Tse and Tsui (2002), and equation (9.108) could also be simplified by specifying A and B each as single scalars so that all the conditional correlations would follow the same process. The model may be estimated in one single stage using maximum likelihood, although this will still be a difficult exercise in the context of large systems. Consequently, Engle advocates a two-stage estimation procedure where each variable in the system is first modelled separately as a univariate GARCH process. A joint log-likelihood function for this stage could be constructed, which would simply be the sum (over N) of all of the log-likelihoods for the individual GARCH models. Then, in the second stage, the conditional likelihood is maximised with respect to any 555

unknown parameters in the correlation matrix. The log-likelihood function for the second stage estimation will be of the form (9.110) where θ1 denotes all the unknown parameters that were estimated in the first stage and θ2 denotes all those to be estimated in the second stage. Estimation using this two-step procedure will be consistent but inefficient as a result of any parameter uncertainty from the first stage being carried through to the second.

9.25 Extensions to the Basic Multivariate GARCH Model Numerous extensions to the univariate specification have been proposed, and many of these carry over to the multivariate case. For example, conditional variance or covariance terms can be included in the conditional mean equation (see Bollerslev, Engle and Wooldridge, 1988, for instance). In the context of financial applications, where the yt are returns, the parameters on these variables can be loosely interpreted as risk premia.

9.25.1 Asymmetric Multivariate GARCH Asymmetric models have become very popular in empirical applications, where the conditional variances and/or covariances are permitted to react differently to positive and negative innovations of the same magnitude. In the multivariate context, this is usually achieved in the Glosten, Jagannathan and Runkle (1993) framework, rather than the EGARCH specification of Nelson (1991). Kroner and Ng (1998), for example, suggest the following extension to the BEKK formulation (with obvious related modifications for the VECH or diagonal VECH models) (9.111) where zt−1 is an N-dimensional column vector with elements taking the value −ϵt−1 if the corresponding element of ϵt−1 is negative and zero otherwise. The asymmetric properties of time-varying covariance matrix models are analysed by Kroner and Ng (1998), who identify three possible 556

forms of asymmetric behaviour. First, the covariance matrix displays own variance asymmetry if the conditional variance of one series is affected by the sign of the innovation in that series. Second, the covariance matrix displays cross variance asymmetry if the conditional variance of one series is affected by the sign of the innovation of another series. Finally, if the conditional covariance is sensitive to the sign of the innovation in return for either series, then the model is said to display covariance asymmetry.

9.25.2 Alternative Distributional Assumptions As was the case for stochastic volatility and univariate GARCH models, an assumption of (multivariate) conditional normality cannot generate sufficiently fat tails to accurately model the distributional properties of financial data. A better approximation to the actual distributions of (especially financial) time series can be obtained using a Student’s t distribution. Such a model can still be estimated using maximum likelihood but with a different (and more complex) likelihood function. The standard formulation will involve estimating, as part of the process, a single degree of freedom parameter which applies to all of the series in the system. An additional potential drawback of this approach is that the tail fatness embodied in the degrees of freedom parameter is fixed over time. Brooks et al. (2005) propose a model where both of these limitations are removed. However, some identifying restrictions are still required. A further issue is the extent to which the unconditional distribution of the shocks is skewed. If this is the case, then a model based on the Student’s t will be inadequate, and an alternative such as the multivariate skew Student’s t of Bauwens and Laurent (2002) must be used. Although many other extensions of the basic models may be conceived of, such as periodic or seasonal MGARCH, the range of specifications employed in the existing literature is narrower than for the corresponding univariate models. A major drawback for even the more parsimonious of the models above is that they are too highly parameterised, and yet many potential applications in economics and finance are in the context of high dimensional systems (such as asset allocation among a number of stocks). Thus, an important innovation was the development of orthogonal and factor models referenced above. Both have the same fundamental idea that by forcing some structure on the variance-covariance matrix, a simplification can be achieved.

9.26 A Multivariate GARCH Model for the CAPM 557

with Time-Varying Covariances Bollerslev, Engle and Wooldridge (1988) estimate a multivariate GARCH model for returns to US Treasury bills, gilts and stocks. The data employed comprised calculated quarterly excess holding period returns for six-month US Treasury bills, twenty-year US Treasury bonds and a Center for Research in Security Prices record of the return on the New York Stock Exchange (NYSE) value-weighted index. The data run from 1959Q1 to 1984Q2 – a total of 102 observations. A multivariate GARCH-M model of the diagonal VECH type is employed, with coefficients estimated by maximum likelihood, and the Berndt et al. (1974) algorithm is used. The coefficient estimates are easiest presented in the following equations for the conditional mean and variance equations, respectively

(9.112)

(9.113)

Source: Bollerslev, Engle and Wooldridge (1988). Reprinted with the permission of University of Chicago Press.

where yjt are the returns, ωjt−1 are a set vector of value weights at time t − 1, i = 1, 2, 3, refers to bills, bonds and stocks, respectively and standard errors are given in parentheses. Consider now the implications of the signs, sizes and significances of the coefficient estimates in equations (9.112) and (9.113). The coefficient of 0.499 in the conditional mean equation gives an aggregate measure of relative risk aversion, also interpreted as representing the market trade-off between return and risk. This conditional 558

variance-in-mean coefficient gives the required additional return as compensation for taking an additional unit of variance (risk). The intercept coefficients in the conditional mean equation for bonds and stocks are very negative and highly statistically significant. The authors argue that this is to be expected since favourable tax treatments for investing in longer-term assets encourages investors to hold them even at relatively low rates of return. The dynamic structure in the conditional variance and covariance equations is strongest for bills and bonds, and very weak for stocks, as indicated by their respective statistical significances. In fact, none of the parameters in the conditional variance or covariance equations for the stock return equations is significant at the 5% level. The unconditional covariance between bills and bonds is positive, while that between bills and stocks, and between bonds and stocks, is negative. This arises since, in the latter two cases, the lagged conditional covariance parameters are negative and larger in absolute value than those of the corresponding lagged error cross-products. Finally, the degree of persistence in the conditional variance (given by α1 + β), which embodies the degree of clustering in volatility, is relatively large for the bills equation, but surprisingly small for bonds and stocks, given the results of other relevant papers in this literature.

9.27 Estimating a Time-Varying Hedge Ratio for FTSE Stock Index Returns A paper by Brooks, Henry and Persand (2002) compared the effectiveness of hedging on the basis of hedge ratios derived from various multivariate GARCH specifications and other, simpler techniques. Some of their main results are discussed below.

9.27.1 Background There has been much empirical research into the calculation of optimal hedge ratios. The general consensus is that the use of multivariate GARCH (MGARCH) models yields superior performances, evidenced by lower portfolio volatilities, than either time-invariant or rolling OLS hedges. Cecchetti, Cumby and Figlewski (1988), Myers and Thompson (1989) and Baillie and Myers (1991), for example, argue that commodity prices are characterised by time-varying covariance matrices. As news about spot 559

and futures prices arrives to the market in discrete bunches, the conditional covariance matrix, and hence the optimal hedging ratio, becomes timevarying. Baillie and Myers (1991) and Kroner and Sultan (1993), inter alia, employ MGARCH models to capture time-variation in the covariance matrix and to estimate the resulting hedge ratio.

9.27.2 Notation Let St and Ft represent the logarithms of the stock index and stock index futures prices, respectively. The actual return on a spot position held from time t − 1 to t is ΔSt = St − St−1 similarly, the actual return on a futures position is ΔFt = Ft − Ft−1. However at time t − 1 the expected return, Et −1(Rt), of the portfolio comprising one unit of the stock index and β units of the futures contract may be written as (9.114) where βt−1 is the hedge ratio determined at time t − 1, for employment in period t. The variance of the expected return, hp,t, of the portfolio may be written as (9.115) where hp,t, hs,t and hF,t represent the conditional variances of the portfolio and the spot and futures positions, respectively and hSF,t represents the conditional covariance between the spot and futures position. the optimal number of futures contracts in the investor’s portfolio, i.e., the optimal hedge ratio, is given by (9.116) If the conditional variance–covariance matrix is time-invariant (and if St and Ft are not cointegrated) then an estimate of β*, the constant optimal hedge ratio, may be obtained from the estimated slope coefficient b in the regression (9.117)

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The OLS estimate of the optimal hedge ratio could be given by b = hSF/hF.

9.27.3 Data and Results The data employed in the Brooks, Henry and Persand (2002) study comprise 3,580 daily observations on the FTSE 100 stock index and stock index futures contract spanning the period 1 January 1985–9 April 1999. Several approaches to estimating the optimal hedge ratio are investigated. The hedging effectiveness is first evaluated in-sample, that is, where the hedges are constructed and evaluated using the same set of data. The outof-sample hedging effectiveness for a one-day hedging horizon is also investigated by forming one-step-ahead forecasts of the conditional variance of the futures series and the conditional covariance between the spot and futures series. These forecasts are then translated into hedge ratios using equation (9.116). The hedging performance of a BEKK formulation is examined, and also a BEKK model including asymmetry terms (in the same style as GJR models). The returns and variances for the various hedging strategies are presented in Table 9.5. Table 9.5 Hedging effectiveness: summary statistics for portfolio returns

Note: t-ratios displayed as {.}. Source: Brooks, Henry and Persand (2002).

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The simplest approach, presented in column (2), is that of no hedge at all. In this case, the portfolio simply comprises a long position in the cash market. Such an approach is able to achieve significant positive returns in sample, but with a large variability of portfolio returns. Although none of the alternative strategies generate returns that are significantly different from zero, either in-sample or out-of-sample, it is clear from columns (3)– (5) of Table 9.5 that any hedge generates significantly less return variability than none at all. The ‘naive’ hedge, which takes one short futures contract for every spot unit, but does not allow the hedge to time-vary, generates a reduction in variance of the order of 80% in-sample and nearly 90% out-of-sample relative to the unhedged position. Allowing the hedge ratio to be timevarying and determined from a symmetric multivariate GARCH model leads to a further reduction as a proportion of the unhedged variance of 5% and 2% for the in-sample and holdout sample, respectively. Allowing for an asymmetric response of the conditional variance to positive and negative shocks yields a very modest reduction in variance (a further 0.5% of the initial value) in-sample, and virtually no change out-of-sample. Figure 9.5 graphs the time-varying hedge ratio from the symmetric and asymmetric MGARCH models (source: Brooks, Henry and Persand, 2002). The optimal hedge ratio is never greater than 0.96 futures contracts per index contract, with an average value of 0.82 futures contracts sold per long index contract. The variance of the estimated optimal hedge ratio is 0.0019. Moreover the optimal hedge ratio series obtained through the estimation of the asymmetric GARCH model appears stationary. An ADF test of the null hypothesis (i.e., that the optimal hedge ratio from the asymmetric BEKK model contains a unit root) was strongly rejected by the data (ADF statistic = −5.7215, 5% Critical value = −2.8630). The timevarying hedge requires the sale (purchase) of fewer futures contracts per long (short) index contract and hence would save the firm wishing to hedge a short exposure money relative to the time-invariant hedge. One possible interpretation of the better performance of the dynamic strategies over the naive hedge is that the dynamic hedge uses short-run information, while the naive hedge is driven by long-run considerations and an assumption that the relationship between spot and futures price movements is 1:1.

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Figure 9.5 Time-varying hedge ratios derived from symmetric and

asymmetric BEKK models for FTSE returns Source. Brooks, Henry and Persand (2002).

Brooks, Henry and Persand also investigate the hedging performances of the various models using a modern risk management approach. They find, once again, that the time-varying hedge results in a considerable improvement, but that allowing for asymmetries results in only a very modest incremental reduction in hedged portfolio risk.

9.28 Multivariate Stochastic Volatility Models As in the univariate case, while the term ‘stochastic volatility’ is commonly used to describe models from the multivariate GARCH family, strictly they do not fit well under this umbrella because the conditional variance and covariance equations are deterministic given the information set up to the previous period. That is, there is no additional source of noise in the conditional variance (or covariance) equation of a multivariate GARCH model. The multivariate stochastic volatility (MSV) model was initially proposed by Harvey, Ruiz and Shephard (1994) and the notation here will 563

closely follow theirs. Let yt be the elements of an N × 1 vector of observations at time t on a series i, with time-varying variance defined as (9.118) where ϵ = (ϵ1t, …, ϵNt) is a vector of disturbances with zero mean and covariance matrix Σϵ and where (9.119) This covariance matrix, Σϵ is defined to have unity on the leading diagonal (and it is therefore also a correlation matrix), while its off-diagonal elements are denoted ρij. Under the stochastic volatility model, the hit can be specified to evolve as an autoregressive (AR) process of order P (9.120) ηt = (η1t, …, ηNt) is a vector of disturbances to the conditional variance having zero mean and covariance matrix Ση. It is usually further assumed that ϵit and ηit are mutually independent and that each is multivariate normally distributed. Usually, P = 1 is deemed sufficient so that the variance dynamics for each series in the system are AR(1). Moving average terms or even exogenous variables could be added to the variance specification but rarely are in practice. It is worth noting that in this model, the correlations ρij between the mean equation disturbances are required to be fixed over time. Thus the covariances across the N series evolve as functions of the variances rather than independently of them. This formulation parallels the constant conditional correlation multivariate GARCH model of Bollerslev (1990) discussed above, and represents an important limitation of the model. It does, however, imply that MSV models are highly parsimonious, and the number of parameters scales directly with the number of variables in the system. For example, in the context of a bivariate MSV model, there are eight parameters to estimate.2 Harvey, Ruiz and Shephard (1994) propose estimating the model using 564

quasimaximum likelihood (QML) via the Kalman filter. However, Danielsson (1998) argues that their QML approach results in inefficient estimation. An alternative approach to estimating MSV models is to make use of Bayesian Markov Chain Monte Carlo (MCMC) methods, as proposed by Jacquier, Polson and Rossi (1995).3 KEY CONCEPTS The key terms to be able to define and explain from this chapter are non-linearity conditional variance maximum likelihood lagrange multiplier test asymmetry in volatility constant conditional correlation diagonal VECH news impact curve volatility clustering GARCH model Wald test likelihood ratio test GJR specification exponentially weighted moving average BEKK model GARCH-in-mean

Appendix 9.1 Parameter Estimation Using Maximum Likelihood For simplicity, this appendix will consider by way of illustration the bivariate regression case with homoscedastic errors (i.e., assuming that there is no ARCH and that the variance of the errors is constant over time). Suppose that the linear regression model of interest is of the form (9A.1) Assuming that ut ~ N(0, σ2), then yt ~ N(β1 + β2xt, σ2) so that the probability density function for a normally distributed random variable 565

with this mean and variance is given by (9A.2) The probability density is a function of the data given the parameters. Successive values of yt would trace out the familiar bell-shaped curve of the normal distribution. Since the ys are iid, the joint probability density function (pdf) for all the ys can be expressed as a product of the individual density functions

(9A.3)

The term on the LHS of this expression is known as the joint density and the terms on the RHS are known as the marginal densities. This result follows from the independence of the y values, in the same way as under elementary probability, for three independent events A, B and C, the probability of A, B and C all happening is the probability of A multiplied by the probability of B multiplied by the probability of C. Equation (9A.3) shows the probability of obtaining all of the values of y that did occur. Substituting into equation (9A.3) for every yt from (9A.2), and using the result that the following expression is obtained

(9A.4)

This is the joint density of all of the ys given the values of xt, β1, β2 and σ2. However, the typical situation that occurs in practice is the reverse of the above situation – that is, the xt and yt are given and β1, β2, σ2 are to be estimated. If this is the case, then f(•) is known as a likelihood function, denoted LF(β1, β2, σ2), which would be written

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(9A.5)

Maximum likelihood estimation involves choosing parameter values (β1, β2, σ2) that maximise this function. Doing this ensures that the values of the parameters are chosen that maximise the likelihood that we would have actually observed the ys that we did. It is necessary to differentiate (9A.5) w.r.t. β1, β2, σ2, but equation (9A.5) is a product containing T terms, and so would be difficult to differentiate. Fortunately, since logs of equation (9A.3) can be taken, and the resulting expression differentiated, knowing that the same optimal values for the parameters will be chosen in both cases. Then, using the various laws for transforming functions containing logarithms, the loglikelihood function, LLF is obtained (9A.6) which is equivalent to (9A.7) Only the first part of the RHS of equation (9A.6) has been changed in equation (9A.7) to make σ2 appear in that part of the expression rather than σ. Remembering the result that

and differentiating equation (9A.7) w.r.t. β1, β2, σ2, the following expressions for the first derivatives are obtained (9A.8) (9A.9) (9A.10)

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Setting equations (9A.8)–(9A.10) to zero to minimise the functions, and placing hats above the parameters to denote the maximum likelihood estimators, from equation (9A.8) (9A.11) (9A.12) (9A.13) (9A.14) Recall that

the mean of y and similarly for x, an estimator for

can finally be derived (9A.15)

From equation (9A.9) (9A.16) (9A.17) (9A.18) (9A.19) (9A.20) (9A.21) (9A.22) From equation (9A.10) (9A.23) Rearranging, (9A.24) But the term in parentheses on the RHS of equation (9A.24) is the residual for time t (i.e., the actual minus the fitted value), so 568

(9A.25) How do these formulae compare with the OLS estimators? Equations (9A.15) and (9A.22) are identical to those of OLS. So maximum likelihood and OLS will deliver identical estimates of the intercept and slope coefficients. However, the estimate of in equation (9A.25) is different. The OLS estimator was (9A.26) and it was also shown that the OLS estimator is unbiased. Therefore, the ML estimator of the error variance must be biased, although it is consistent, since as T → ∞, T − k ≈ T. Note that the derivation above could also have been conducted using matrix rather than sigma algebra. The resulting estimators for the intercept and slope coefficients would still be identical to those of OLS, while the estimate of the error variance would again be biased. It is also worth noting that the ML estimator is consistent and asymptotically efficient. Derivation of the ML estimator for the GARCH LLF is algebraically difficult and therefore beyond the scope of this book. SELF-STUDY QUESTIONS 1. (a) What stylised features of financial data cannot be explained using linear time series models? (b) Which of these features could be modelled using a GARCH(1,1) process? (c) Why, in recent empirical research, have researchers preferred GARCH(1,1) models to pure ARCH(p)? (d) Describe two extensions to the original GARCH model. What additional characteristics of financial data might they be able to capture? (e) Consider the following GARCH(1,1) model

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If yt is a daily stock return series, what range of values are likely for the coefficients μ, α0, α1 and β? (f) Suppose that a researcher wanted to test the null hypothesis that α1 + β = 1 in the equation for part (e). Explain how this might be achieved within the maximum likelihood framework. (g) Suppose now that the researcher had estimated the above GARCH model for a series of returns on a stock index and obtained the following parameter estimates: If the researcher has data available up to and including time T, write down a set of equations in and their lagged values, which could be employed to produce one-, two-, and three-step-ahead forecasts for the conditional variance of yt. (h) Suppose now that the coefficient estimate of for this model is 0.98 instead. By reconsidering the forecast expressions you derived in part (g), explain what would happen to the forecasts in this case. 2. (a) Discuss briefly the principles behind maximum likelihood. (b) Describe briefly the three hypothesis testing procedures that are available under maximum likelihood estimation. Which is likely to be the easiest to calculate in practice, and why? (c) OLS and maximum likelihood are used to estimate the parameters of a standard linear regression model. Will they give the same estimates? Explain your answer. 3. (a) Distinguish between the terms ‘conditional variance’ and ‘unconditional variance’. Which of the two is more likely to be relevant for producing: i. one-step-ahead volatility forecasts ii. twenty-step-ahead volatility forecasts. (b) If ut follows a GARCH(1,1) process, what would be the likely result if a regression of the form in Question 1(e) were estimated using OLS and assuming a constant conditional variance? (c) Compare and contrast the following models for volatility, noting their strengths and weaknesses: 570

i. Historical volatility ii. EWMA iii. GARCH(1,1) iv. Implied volatility. 4. Suppose that a researcher is interested in modelling the correlation between the returns of the NYSE and LSE markets. (a) Write down a simple diagonal VECH model for this problem. Discuss the values for the coefficient estimates that you would expect. (b) Suppose that weekly correlation forecasts for two weeks ahead are required. Describe a procedure for constructing such forecasts from a set of daily returns data for the two market indices. (c) What other approaches to correlation modelling are available? (d) What are the strengths and weaknesses of multivariate GARCH models relative to the alternatives that you propose in part (c)? 5. (a) What is a news impact curve? Using a spreadsheet or otherwise, construct the news impact curve for the following estimated EGARCH and GARCH models, setting the lagged conditional variance to the value of the unconditional variance (estimated from the sample data rather than the mode parameter estimates), which is 0.096

GARCH EGARCH μ −0.0130 −0.0278 (0.0669) (0.0855) α0 0.0019 0.0823 α1

(0.0017) (0.5728) 0.1022** −0.0214 (0.0333) (0.0332) 571

α2 0.9050** 0.9639** α3

(0.0175) (0.0136) − 0.2326** (0.0795)

(b)

1 2 3

In fact, the models in part (a) were estimated using daily foreign exchange returns. How can financial theory explain the patterns observed in the news impact curves?

The BEKK acronym arises from the fact that early versions of the paper also listed Baba and Krafts as co-authors. This compares with nine for a diagonal VECH MGARCH model and 21 for the unrestricted MGARCH. See Chib and Greenberg (1996) for an extensive but very technical discussion of the intricacies of the MCMC technique.

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10 Switching and State Space Models

LEARNING OUTCOMES In this chapter, you will learn how to Use intercept and slope dummy variables to allow for seasonal behaviour in time-series Motivate the use of regime switching models in financial econometrics Specify and explain the logic behind Markov switching models Compare and contrast Markov switching and threshold autoregressive models Describe the intuition behind the estimation of regime switching models Set up and interpret simple state space models Explain how the Kalman filter is used to estimate state space models

10.1 Motivations Many financial and economic time series seem to undergo episodes in which the behaviour of the series changes quite dramatically compared to that exhibited previously. The behaviour of a series could change over time in terms of its mean value, its volatility, or to what extent its current value is related to its previous value. The behaviour may change once and for all, usually known as a ‘structural break’ in a series. Or it may change for a period of time before reverting back to its original behaviour or switching to yet another style of behaviour, and the latter is typically 573

termed a ‘regime shift’ or ‘regime switch’. Finally, the relationship between series or their average values may change continuously but in an at least partially predictable fashion. This chapter presents several models that can capture such time-varying behaviour.

10.1.1 What Might Cause One-Off Fundamental Changes in the Properties of a Series? Usually, very substantial changes in the properties of a series are attributed to large-scale events, such as wars, financial panics – e.g., a ‘run on a bank’, significant changes in government policy, such as the introduction of an inflation target, or the removal of exchange controls, or changes in market microstructure – e.g., the ‘Big Bang’, when trading on the London Stock Exchange (LSE) became electronic, or a change in the market trading mechanism, such as the partial move of the LSE from a quotedriven to an order-driven system in 1997. However, it is also true that regime shifts can occur on a regular basis and at much higher frequency. Such changes may occur as a result of more subtle factors, but still leading to statistically important modifications in behaviour. An example would be the intraday patterns observed in equity market bid–ask spreads (see Chapter 7). These appear to start with high values at the open, gradually narrowing throughout the day, before widening again at the close. To give an illustration of the kind of shifts that may be seen to occur, Figure 10.1 gives an extreme example. As can be seen from the figure, the behaviour of the series changes markedly at around observation 500. Not only does the series become much more volatile than previously, its mean value is also substantially increased. Although this is a severe case that was generated using simulated data, clearly, in the face of such ‘regime changes’ a linear model estimated over the whole sample covering the change would not be appropriate. One possible approach to this problem would be simply to split the data around the time of the change and to estimate separate models on each portion. It would be possible to allow a series, yt to be drawn from two or more different generating processes at different times. For example, if it was thought an AR(1) process was appropriate to capture the relevant features of a particular series whose behaviour changed at observation 500, say, two models could be estimated: (10.1) 574

(10.2)

Figure 10.1 Sample time-series plot illustrating a regime shift

In the context of Figure 10.1, this would involve focusing on the mean shift only. These equations represent a very simple example of what is known as a piecewise linear model – that is, although the model is globally (i.e., when it is taken as a whole) non-linear, each of the component parts is a linear model. This method may be valid, but it is also likely to be wasteful of information. For example, even if there were enough observations in each sub-sample to estimate separate (linear) models, there would be an efficiency loss in having fewer observations in each of two samples than if all the observations were collected together. Also, it may be the case that only one property of the series has changed – for example, the (unconditional) mean value of the series may have changed, leaving its other properties unaffected. In this case, it would be sensible to try to keep all of the observations together, but to allow for the particular form of the structural change in the model-building process. Thus, what is required is a set of models that allow all of the observations on a series to be used for estimating a model, but also that the model is sufficiently flexible to allow different types of behaviour at different points in time. Two classes of regime switching models that potentially allow this to occur are Markov switching models and threshold autoregressive models. A first and central question to ask is: How can it be determined where 575

the switch(es) occurs? The method employed for making this choice will depend upon the model used. A simple type of switching model is one where the switches are made deterministically using dummy variables. One important use of this in finance is to allow for ‘seasonality’ in financial data. In economics and finance generally, many series are believed to exhibit seasonal behaviour, which results in a certain element of partly predictable cycling of the series over time. For example, if monthly or quarterly data on consumer spending are examined, it is likely that the value of the series will rise rapidly in late November owing to Christmas-related expenditure, followed by a fall in mid-January, when consumers realise that they have spent too much before Christmas and in the January sales! Consumer spending in the UK also typically drops during the August vacation period when all of the sensible people have left the country. Such phenomena will be apparent in many series and will be present to some degree at the same time every year, whatever else is happening in terms of the long-term trend and short-term variability of the series.

10.2 Seasonalities in Financial Markets: Introduction and Literature Review In the context of financial markets, and especially in the case of equities, a number of other ‘seasonal effects’ have been noted. Such effects are usually known as ‘calendar anomalies’ or ‘calendar effects’. Examples include open- and close-of-market effects, ‘the January effect’, weekend effects and bank holiday effects. Investigation into the existence or otherwise of ‘calendar effects’ in financial markets has been the subject of a considerable amount of recent academic research. Calendar effects may be loosely defined as the tendency of financial asset returns to display systematic patterns at certain times of the day, week, month or year. One example of the most important such anomalies is the day-of-the-week effect, which results in average returns being significantly higher on some days of the week than others. Studies by French (1980), Gibbons and Hess (1981) and Keim and Stambaugh (1984), for example, have found that the average market close-to-close return in the US is significantly negative on Monday and significantly positive on Friday. By contrast, Jaffe and Westerfield (1985) found that the lowest mean returns for the Japanese and Australian stock markets occur on Tuesdays. At first glance, these results seem to contradict the efficient markets 576

hypothesis, since the existence of calendar anomalies might be taken to imply that investors could develop trading strategies which make abnormal profits on the basis of such patterns. For example, holding all other factors constant, equity purchasers may wish to sell at the close on Friday and to buy at the close on Thursday in order to take advantage of these effects. However, evidence for the predictability of stock returns does not necessarily imply market inefficiency, for at least two reasons. First, it is likely that the small average excess returns documented by the above papers would not generate net gains when employed in a trading strategy once the costs of transacting in the markets has been taken into account. Therefore, under many ‘modern’ definitions of market efficiency (e.g., Jensen, 1978), these markets would not be classified as inefficient. Second, the apparent differences in returns on different days of the week may be attributable to time-varying stock market risk premiums. If any of these calendar phenomena are present in the data but ignored by the model-building process, the result is likely to be a misspecified model. For example, ignored seasonality in yt is likely to lead to residual autocorrelation of the order of the seasonality – e.g., fifth order residual autocorrelation if yt is a series of daily returns.

10.3 Modelling Seasonality in Financial Data As discussed above, seasonalities at various different frequencies in financial time-series data are so well documented that their existence cannot be doubted, even if there is argument about how they can be rationalised. One very simple method for coping with this and examining the degree to which seasonality is present is the inclusion of dummy variables in regression equations. The number of dummy variables that could sensibly be constructed to model the seasonality would depend on the frequency of the data. For example, four dummy variables would be created for quarterly data, twelve for monthly data, five for daily data and so on. In the case of quarterly data, the four dummy variables would be defined as follows: D1t = 1 in quarter 1 and zero otherwise D2t = 1 in quarter 2 and zero otherwise D3t = 1 in quarter 3 and zero otherwise D4t = 1 in quarter 4 and zero otherwise 577

How many dummy variables can be placed in a regression model? If an intercept term is used in the regression, the number of dummies that could also be included would be one less than the ‘seasonality’ of the data. To see why this is the case, consider what happens if all four dummies are used for the quarterly series. The following gives the values that the dummy variables would take for a period during the mid-1980s, together with the sum of the dummies at each point in time, presented in the last column

1986 Q1 Q2 Q3 Q4 1987 Q1 Q2 Q3

D1 D2 D3 D4 Sum 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 etc.,

The sum of the four dummies would be 1 in every time period. Unfortunately, this sum is of course identical to the variable that is implicitly attached to the intercept coefficient. Thus, if the four dummy variables and the intercept were both included in the same regression, the problem would be one of perfect multicollinearity so that (X′X)−1 would not exist and none of the coefficients could be estimated. This problem is known as the dummy variable trap. The solution would be either to just use three dummy variables plus the intercept, or to use the four dummy variables with no intercept. The seasonal features in the data would be captured using either of these, and the residuals in each case would be identical, although the interpretation of the coefficients would be changed. If four dummy variables were used (and assuming that there were no explanatory variables in the regression), the estimated coefficients could be interpreted as the average value of the dependent variable during each quarter. In the case where a constant and three dummy variables were used, the interpretation of the estimated coefficients on the dummy variables would be that they represented the average deviations of the dependent variables for the included quarters from their average values for the excluded 578

quarter, as discussed in Box 10.1. BOX 10.1 How do dummy variables work? The dummy variables as described above operate by changing the intercept, so that the average value of the dependent variable, given all of the explanatory variables, is permitted to change across the seasons. This is shown in Figure 10.2.

Figure 10.2 Use of intercept dummy variables for quarterly data

Consider the following regression (10.3) During each period, the intercept will be changed. The intercept will be: in the first quarter, since D1 = 1 and D2 = D3 = 0 for all quarter 1 observations in the second quarter, since D2 = 1 and D1 = D3 = 0 for all quarter 2 observations. 579

in the third quarter, since D3 = 1 and D1 = D2 = 0 for all quarter 3 observations in the fourth quarter, since D1 = D2 = D3 = 0 for all quarter 4 observations.

EXAMPLE 10.1 Brooks and Persand (2001a) examine the evidence for a day-of-theweek effect in five Southeast Asian stock markets: South Korea, Malaysia, the Philippines, Taiwan and Thailand. The data, obtained from Primark Datastream, are collected on a daily close-to-close basis for all weekdays (Mondays to Fridays) falling in the period 31 December 1989 to 19 January 1996 (a total of 1,581 observations). The first regressions estimated, which constitute the simplest tests for dayof-the-week effects, are of the form (10.4) where rt is the return at time t for each country examined separately, D1t is a dummy variable for Monday, taking the value 1 for all Monday observations and zero otherwise, and so on. The coefficient estimates can be interpreted as the average sample return on each day of the week. The results from these regressions are shown in Table 10.1. Table 10.1 Values and significances of days of the week coefficients

Monday Tuesday Wednesday Thursday

Thailand

Malaysia

0.49E-3

0.00322

0.00185

0.56E-3

0.00119

(0.6740)

(3.9804)**

(2.9304)**

(0.4321)

(1.4369)

−0.45E-3

−0.00179

−0.00175

0.00104

−0.97E-4

(−0.3692)

(−1.6834)

(−2.1258)**

(0.5955)

(−0.0916)

−0.37E-3

−0.00160

0.31E-3

−0.00264

−0.49E-3

(−0.5005)

(−1.5912)

(0.4786)

(−2.107)**

(−0.5637)

0.40E-3

0.00100

0.00159

−0.00159

0.92E-3

580

Taiwan

South Korea

Philippines

Friday

(0.5468)

(1.0379)

(2.2886)**

(−1.2724)

(0.8908)

−0.31E-3

0.52E-3

0.40E-4

0.43E-3

0.00151

(−0.3998)

(0.5036)

(0.0536)

(0.3123)

(1.7123)

Notes: Coefficients are given in each cell followed by t-ratios in parentheses; and** denote significance at the 5% and 1% levels, respectively. Source: Brooks and Persand (2001a).

*

Briefly, the main features are as follows. Neither Thailand nor the Philippines have significant calendar effects; both Taiwan and Malaysia have significant positive Monday average returns and significant negative Tuesday returns; Taiwan has a significant Thursday effect. Dummy variables could also be used to test for other calendar anomalies, such as the January effect, etc. as discussed above, and a given regression can include dummies of different frequencies at the same time. For example, a new dummy variable D6t could be added to equation (10.4) for ‘April effects’, associated with the start of the new tax year in the UK. Such a variable, even for a regression using daily data, would take the value 1 for all observations falling in April and zero otherwise. If we choose to omit one of the dummy variables and to retain the intercept, then the omitted dummy variable becomes the reference category against which all the others are compared. For example consider a model such as the one above, but where the Monday dummy variable has been omitted (10.5) The estimate of the intercept will be on Monday, on Tuesday and so on. will now be interpreted as the difference in average returns between Monday and Tuesday. Similarly, can also be interpreted as the differences in average returns between Wednesday, …, Friday, and Monday. This analysis should hopefully have made it clear that by thinking carefully about which dummy variable (or the intercept) to omit from the regression, we can control the interpretation to test naturally the hypothesis that is of most interest. The same logic can also be applied to slope dummy variables, which are described in the following 581

section.

10.3.1 Slope Dummy Variables As well as, or instead of, intercept dummies, slope dummy variables can also be used. These operate by changing the slope of the regression line, leaving the intercept unchanged. Figure 10.3 gives an illustration in the context of just one slope dummy (i.e., two different ‘states’). Such a setup would apply if, for example, the data were bi-annual (twice yearly) or biweekly or observations made at the open and close of markets. Then Dt would be defined as Dt = 1 for the first half of the year and zero for the second half.

Figure 10.3 Use of slope dummy variables

A slope dummy changes the slope of the regression line, leaving the intercept unchanged. In the above case, the intercept is fixed at α, while the slope varies over time. For periods where the value of the dummy is zero, the slope will be β, while for periods where the dummy is one, the slope will be β + γ. Of course, it is also possible to use more than one dummy variable for the slopes. For example, if the data were quarterly, the following setup could be used, with D1t … D3t representing quarters 1–3

582

(10.6) In this case, since there is also a term in xt with no dummy attached, the interpretation of the coefficients on the dummies (γ1, etc.) is that they represent the deviation of the slope for that quarter from the average slope over all quarters. On the other hand, if the four slope dummy variables were included (and not βxt), the coefficients on the dummies would be interpreted as the average slope coefficients during each quarter. Again, it is important not to include four quarterly slope dummies and the βxt in the regression together, otherwise perfect multicollinearity would result.

10.3.2 Interactive Dummy Variables It is often of interest to examine how variables interact with one another. This is achieved in a regression model by including them multiplied together. Frequently, a dummy variable will be interacted either with another dummy variable or with a standard explanatory variable. In the following example, we show how a seasonal dummy variable can be interacted with the market risk factor to allow for time-varying risk. To offer an illustration of how two dummy variables can be interacted together, consider a regression model where we are trying to explain the risk levels (riski) of the portfolios of professional fund managers according to the manager’s age and their gender (10.7) where DGi is a gender dummy variable, taking the value 1 if the fund manager is female and 0 otherwise; DAi is a dummy variable for the age of the fund manager taking the value 0 for less than 40 years old and 1 otherwise. We would, of course, usually add some additional explanatory variables to the model but none are included here for simplicity. EXAMPLE 10.2 Returning to the example of day-of-the-week effects in Southeast Asian stock markets, although significant coefficients in equation (10.4) will support the hypothesis of seasonality in returns, it is important to note that risk factors have not been taken into account. Before drawing conclusions on the potential presence of arbitrage opportunities or 583

inefficient markets, it is important to allow for the possibility that the market can be more or less risky on certain days than others. Hence, low (high) significant returns in equation (10.4) might be explained by low (high) risk. Brooks and Persand thus test for seasonality using the empirical market model, whereby market risk is proxied by the return on the FTA World Price Index. Hence, in order to look at how risk varies across the days of the week, interactive (i.e., slope) dummy variables are used to determine whether risk increases (decreases) on the day of high (low) returns. The equation, estimated separately using time-series data for each country can be written (10.8) where αi and βi are coefficients to be estimated, Dit is the ith dummy variable taking the value 1 for day t = i and zero otherwise, and RWMt is the return on the world market index. In this way, when considering the effect of market risk on seasonality, both risk and return are permitted to vary across the days of the week. The results from estimation of equation (10.8) are given in Table 10.2. Note that South Korea and the Philippines are excluded from this part of the analysis, since no significant calendar anomalies were found to explain in Table 10.1. Table 10.2 Day-of-the-week effects with the inclusion of interactive dummy variables with the risk proxy

Monday Tuesday Wednesday Thursday

Thailand

Malaysia

0.00322

0.00185

0.544E-3

(3.3571)**

(2.8025)**

(0.3945)

−0.00114

−0.00122

0.00140

(−1.1545)

(−1.8172)

(1.0163)

−0.00164

0.25E-3

−0.00263

(−1.6926)

(0.3711)

(−1.9188)

0.00104

0.00157

−0.00166

(1.0913)

(2.3515)*

(−1.2116)

584

Taiwan

Friday BetaMonday

BetaTuesday BetaWednesday BetaThursday Beta-Friday

0.31E-4

−0.3752

−0.13E-3

(0.03214)

(−0.5680)

(−0.0976)

0.3573

0.5494

0.6330

(2.1987)*

(4.9284)**

(2.7464)**

1.0254

0.9822

0.6572

(8.0035)**

(11.2708)**

(3.7078)**

0.6040

0.5753

0.3444

(3.7147)**

(5.1870)**

(1.4856)

0.6662

0.8163

0.6055

(3.9313)**

(6.9846)**

(2.5146)*

0.9124

0.8059

1.0906

(5.8301)**

(7.4493)**

(4.9294)**

Notes: Coefficients are given in each cell followed by t-ratios in parentheses;* and** denote significance at the 5% and 1%, levels respectively. Source: Brooks and Persand (2001a).

As can be seen, significant Monday effects in the Bangkok and Kuala Lumpur stock exchanges, and a significant Thursday effect in the latter, remain even after the inclusion of the slope dummy variables which allow risk to vary across the week. The t-ratios do fall slightly in absolute value, however, indicating that the day-of-the-week effects become slightly less pronounced. The significant negative average return for the Taiwanese stock exchange, however, completely disappears. It is also clear that average risk levels vary across the days of the week. For example, the betas for the Bangkok stock exchange vary from a low of 0.36 on Monday to a high of over unity on Tuesday. This illustrates that not only is there a significant positive Monday effect in this market, but also that the responsiveness of Bangkok market movements to changes in the value of the general world stock market is considerably lower on this day than on other days of the week. 585

It is evident in equation (10.7) that the age and gender dummy variables are included both as individual terms and also interacted with each other. This allows increasing age to have a different affect on the amount of risk taken by a man compared with a woman. Some possible outcomes and their interpretations are is significant but and are insignificant. This would suggest that there is a statistical difference between the average risk levels of men and women that does not vary with age but there is no difference between the risk levels of older and younger fund managers. and are significant but is insignificant. This would suggest that there is a statistical difference between the risk levels of male and female fund managers, and between those who are younger and older, but the differences in the amount of risk taken by male and female managers does not vary with age (or put differently but equivalently, the differences in the amount of risk taken by younger versus older fund managers does not vary by gender). The other possible outcomes would be interpreted similarly. If we assume that all three parameter estimates are non-zero, we would calculate the average levels of risk taken by each group of fund managers as follows (DGi = 0 and DAi = 0, so picking out younger men) (DGi = 1 and DAi = 0, so picking out younger women) (DGi = 0 and DAi = 1, so picking out older men) (DGi = 1 and DAi = 1, so picking out older women)

10.4 Estimating Simple Piecewise Linear Functions The piecewise linear model is one example of a general set of models known as spline techniques. Spline techniques involve the application of polynomial functions in a piecewise fashion to different portions of the data. These models are widely used to fit yield curves to available data on the yields of bonds of different maturities (see, for example, Shea, 1984). A simple piecewise linear model could operate as follows. If the relationship between two series, y and x, differs depending on whether x is smaller or larger than some threshold value x*, this phenomenon can be 586

captured using dummy variables. A dummy variable, Dt, could be defined, taking values (10.9) To offer an illustration of where this may be useful, it is sometimes the case that the tick size limits vary according to the price of the asset. For example, according to George and Longstaff (1993, see also Chapter 6 of this book), the Chicago Board of Options Exchange (CBOE) limits the tick size to be $(1/8) for options worth $3 or more, and $(1/16) for options worth less than $3. This means that the minimum permissible price movements are $(1/8) and ($1/16) for options worth $3 or more and less than $3, respectively. Thus, if y is the bid–ask spread for the option, and x is the option price, used as a variable to partly explain the size of the spread, the spread will vary with the option price partly in a piecewise manner owing to the tick size limit. The model could thus be specified as (10.10) with Dt defined as above. Viewed in the light of the above discussion on seasonal dummy variables, the dummy in equation (10.10) is used as both an intercept and a slope dummy. An example showing the data and regression line is given by Figure 10.4.

Figure 10.4 Piecewise linear model with threshold x*

587

Note that the value of the threshold or ‘knot’ is assumed known at this stage.1 Throughout, it is also possible that this situation could be generalised to the case where yt is drawn from more than two regimes or is generated by a more complex model.

10.5 Markov Switching Models Although a large number of more complex, non-linear threshold models have been proposed in the econometrics literature, only two kinds of model have had any noticeable impact in finance (aside from threshold GARCH models of the type alluded to in Chapter 8). These are the Markov regime switching model associated with Hamilton (1989, 1990), and the threshold autoregressive model associated with Tong (1983, 1990). Each of these formulations will be discussed below.

10.5.1 Fundamentals of Markov Switching Models Under the Markov switching approach, the universe of possible occurrences is split into m states of the world, denoted si, i = 1, …, m, corresponding to m regimes. In other words, it is assumed that yt switches regime according to some unobserved variable, st, that takes on integer values. In the remainder of this chapter, it will be assumed that m = 1 or 2. So if st = 1, the process is in regime 1 at time t, and if st = 2, the process is in regime 2 at time t. Movements of the state variable between regimes are governed by a Markov process. This Markov property can be expressed as (10.11) In plain English, this equation states that the probability distribution of the state at any time t depends only on the state at time t − 1 and not on the states that were passed through at times t − 2, t − 3, …Hence Markov processes are not path-dependent. The model’s strength lies in its flexibility, being capable of capturing changes in the variance between state processes, as well as changes in the mean. The most basic form of Hamilton’s model, also known as ‘Hamilton’s filter’ (see Hamilton, 1989), comprises an unobserved state variable, denoted zt, that is postulated to evaluate according to a first order Markov process

588

(10.12) (10.13) (10.14) (10.15) where p11 and p22 denote the probability of being in regime 1, given that the system was in regime 1 during the previous period, and the probability of being in regime 2, given that the system was in regime 2 during the previous period, respectively. Thus 1−p11 defines the probability that yt will change from state 1 in period t −1 to state 2 in period t, and 1 − p22 defines the probability of a shift from state 2 to state 1 between times t − 1 and t. It can be shown that under this specification, zt evolves as an AR(1) process (10.16) where ρ = p11 + p22 − 1. Loosely speaking, zt can be viewed as a generalisation of the dummy variables for one-off shifts in a series discussed above. Under the Markov switching approach, there can be multiple shifts from one set of behaviour to another. In this framework, the observed returns series evolves as given by equation (10.17) (10.17) where ut ~ N(0, 1). The expected values and variances of the series are μ1 and respectively in state 1, and (μ1 + μ2) and inrespectively, state 2. The variance in state 2 is also defined, The unknown parameters of the model (μ1, μ2, p11, p22) are estimated using maximum likelihood. Details are beyond the scope of this book, but are most comprehensively given in Engel and Hamilton (1990). If a variable follows a Markov process, all that is required to forecast the probability that it will be in a given regime during the next period is the current period’s probability and a set of transition probabilities, given for the case of two regimes by equations (10.12)–(10.15). In the general case where there are m states, the transition probabilities are best expressed in a matrix as

589

(10.18)

where Pij is the probability of moving from regime i to regime j. Since, at any given time, the variable must be in one of the m states, it must be true that

(10.19)

A vector of current state probabilities is then defined as (10.20) where πi is the probability that the variable y is currently in state i. Given πt and P, the probability that the variable y will be in a given regime next period can be forecast using (10.21) The probabilities for S steps into the future will be given by (10.22)

10.6 A Markov Switching Model for the Real Exchange Rate There have been a number of applications of the Markov switching model in finance. Clearly, such an approach is useful when a series is thought to undergo shifts from one type of behaviour to another and back again, but where the ‘forcing variable’ that causes the regime shifts is unobservable. One such application is to modelling the real exchange rate. As discussed in Chapter 8, purchasing power parity (PPP) theory suggests that the law of one price should always apply in the long run such that the cost of a representative basket of goods and services is the same wherever it is 590

purchased, after converting it into a common currency. Under some assumptions, one implication of PPP is that the real exchange rate (that is, the exchange rate divided by a general price index such as the consumer price index (CPI)) should be stationary. However, a number of studies have failed to reject the unit root null hypothesis in real exchange rates, indicating evidence against the PPP theory. It is widely known that the power of unit root tests is low in the presence of structural breaks as the ADF test finds it difficult to distinguish between a stationary process subject to structural breaks and a unit root process. In order to investigate this possibility, Bergman and Hansson (2005) estimate a Markov switching model with an AR(1) structure for the real exchange rate, which allows for multiple switches between two regimes. The specification they use is (10.23) where yt is the real exchange rate, st, (t = 1, 2) are the two states, and ϵt ~ N (0, σ2).2 The state variable st is assumed to follow a standard 2-regime Markov process as described above. Quarterly observations from 1973Q2 to 1997Q4 (99 data points) are used on the real exchange rate (in units of foreign currency per US dollar) for the UK, France, Germany, Switzerland, Canada and Japan. The model is estimated using the first seventy-two observations (1973Q2–1990Q4) with the remainder retained for out-of-sample forecast evaluation. The authors use 100 times the log of the real exchange rate, and this is normalised to take a value of one for 1973Q2 for all countries. The Markov switching model estimates obtained using maximum likelihood estimation are presented in Table 10.3. Table 10.3 Estimates of the Markov switching model for real exchange rates

Note: Standard errors in parentheses. Source: Bergman and Hansson (2005).

591

Reprinted with the permission of Elsevier.

As Table 10.3 shows, the model is able to separate the real exchange rates into two distinct regimes for each series, with the intercept in regime 1 (μ1) being positive for all countries except Japan (resulting from the phenomenal strength of the yen over the sample period), corresponding to a rise in the log of the number of units of the foreign currency per US dollar, i.e., a depreciation of the domestic currency against the dollar. μ2, the intercept in regime 2, is negative for all countries, corresponding to a domestic currency appreciation against the dollar. The probabilities of remaining within the same regime during the following period (p11 and p22) are fairly low for the UK, France, Germany and Switzerland, indicating fairly frequent switches from one regime to another for those countries’ currencies. Interestingly, after allowing for the switching intercepts across the regimes, the AR(1) coefficient, ϕ, in Table 10.3 is a considerable distance below unity, indicating that these real exchange rates are stationary. Bergman and Hansson simulate data from the stationary Markov switching AR(1) model with the estimated parameters but they assume that the researcher conducts a standard ADF test on the artificial data. They find that for none of the cases can the unit root null hypothesis be rejected, even though clearly this null is wrong as the simulated data are stationary. It is concluded that a failure to account for time-varying intercepts (i.e., structural breaks) in previous empirical studies on real exchange rates could have been the reason for the finding that the series are unit root processes when the financial theory had suggested that they should be stationary. Finally, the authors employ their Markov switching AR(1) model for forecasting the remainder of the exchange rates in the sample in comparison with the predictions produced by a random walk and by a Markov switching model with a random walk. They find that for all six series, and for forecast horizons up to four steps (quarters) ahead, their Markov switching AR model produces predictions with the lowest mean squared errors; these improvements over the pure random walk are statistically significant.

10.7 A Markov Switching Model for the Gilt–Equity Yield Ratio 592

As discussed below, a Markov switching approach is also useful for modelling the time series behaviour of the gilt–equity yield ratio (GEYR), defined as the ratio of the income yield on long-term government bonds to the dividend yield on equities. It has been suggested that the current value of the GEYR might be a useful tool for investment managers or market analysts in determining whether to invest in equities or whether to invest in gilts. Thus the GEYR is purported to contain information useful for determining the likely direction of future equity market trends. The GEYR is assumed to have a long-run equilibrium level, deviations from which are taken to signal that equity prices are at an unsustainable level. If the GEYR becomes high relative to its long-run level, equities are viewed as being expensive relative to bonds. The expectation, then, is that for given levels of bond yields, equity yields must rise, which will occur via a fall in equity prices. Similarly, if the GEYR is well below its long-run level, bonds are considered expensive relative to stocks, and by the same analysis, the price of the latter is expected to increase. Thus, in its crudest form, an equity trading rule based on the GEYR would say, ‘if the GEYR is low, buy equities; if the GEYR is high, sell equities’. The paper by Brooks and Persand (2001b) discusses the usefulness of the Markov switching approach in this context, and considers whether profitable trading rules can be developed on the basis of forecasts derived from the model. Brooks and Persand (2001b) employ monthly stock index dividend yields and income yields on government bonds covering the period January 1975 until August 1997 (272 observations) for three countries – the UK, the US and Germany. The series used are the dividend yield and index values of the FTSE100 (UK), the S&P500 (US) and the DAX (Germany). The bond indices and redemption yields are based on the clean prices of UK government consols, and US and German ten-year government bonds. As an example, Figure 10.5 presents a plot of the distribution of the GEYR for the US (in blue), together with a normal distribution having the same mean and variance (source: Brooks and Persand, 2001b). Clearly, the distribution of the GEYR series is not normal, and the shape suggests two separate modes: one upper part of the distribution embodying most of the observations, and a lower part covering the smallest values of the GEYR.

593

Figure 10.5 Unconditional distribution of US GEYR together with a

normal distribution with the same mean and variance Such an observation, together with the notion that a trading rule should be developed on the basis of whether the GEYR is ‘high’ or ‘low’, and in the absence of a formal econometric model for the GEYR, suggests that a Markov switching approach may be useful. Under the Markov switching approach, the values of the GEYR are drawn from a mixture of normal distributions, where the weights attached to each distribution sum to one and where movements between series are governed by a Markov process. The Markov switching model is estimated using a maximum likelihood procedure (as discussed in Chapter 9), based on GAUSS code supplied by James Hamilton. Coefficient estimates for the model are presented in Table 10.4. Table 10.4 Estimated parameters for the Markov switching models

594

Notes: Standard errors in parentheses; N1 and N2 denote the number of observations deemed to be in regimes 1 and 2, respectively. Source: Brooks and Persand (2001b).

The means and variances for the values of the GEYR for each of the two regimes are given in columns headed (1)–(4) of Table 10.4 with standard errors associated with each parameter in parentheses. It is clear that the regime switching model has split the data into two distinct samples – one with a high mean (of 2.43, 2.46 and 3.03 for the UK, US and Germany, respectively) and one with a lower mean (of 2.07, 2.12, and 2.16), as was anticipated from the unconditional distribution of returns. Also apparent is the fact that the UK and German GEYR are more variable at times when it is in the high mean regime, evidenced by their higher variance (in fact, it is around four and twenty times higher than for the low GEYR state, respectively). The number of observations for which the probability that the GEYR is in the high mean state exceeds 0.5 (and thus when the GEYR is actually deemed to be in this state) is 102 for the UK (37.5% of the total), while the figures for the US are 100 (36.8%) and for Germany 200 (73.5%). Thus, overall, the GEYR is more likely to be in the low mean regime for the UK and US, while it is likely to be high in Germany. The columns marked (5) and (6) of Table 10.4 give the values of p11 and p22, respectively, that is the probability of staying in state 1 given that the GEYR was in state 1 in the immediately preceding month, and the probability of staying in state 2 given that the GEYR was in state 2 previously, respectively. The high values of these parameters indicates that the regimes are highly stable with less than a 10% chance of moving from a low GEYR to a high GEYR regime and vice versa for all three series. Figure 10.6 presents a ‘q-plot’, which shows the value of GEYR and probability that it is in the high GEYR regime for the UK at each point in time (source: Brooks and Persand, 2001b). 595

Figure 10.6 Value of GEYR and probability that it is in the high GEYR

regime for the UK As can be seen, the probability that the UK GEYR is in the ‘high’ regime (the dotted line) varies frequently, but spends most of its time either close to zero or close to one. The model also seems to do a reasonably good job of specifying which regime the UK GEYR should be in, given that the probability seems to match the broad trends in the actual GEYR (the full line). Engel and Hamilton (1990) show that it is possible to give a forecast of the probability that a series yt, which follows a Markov switching process, will be in a particular regime. Brooks and Persand (2001b) use the first sixty observations (January 1975–December 1979) for in-sample estimation of the model parameters (μ1, μ2, p11, p22). Then a one stepahead forecast is produced of the probability that the GEYR will be in the high mean regime during the next period. If the probability that the GEYR will be in the low regime during the next period is forecast to be more that 0.5, it is forecast that the GEYR will be low and hence equities are bought or held. If the probability that the GEYR is in the low regime is forecast to be less than 0.5, it is anticipated that the GEYR will be high and hence gilts are invested in or held. The model is then rolled forward one observation, with a new set of model parameters and probability forecasts 596

being constructed. This process continues until 212 such probabilities are estimated with corresponding trading rules. The returns for each out-of-sample month for the switching portfolio are calculated, and their characteristics compared with those of buy-and-hold equities and buy-and-hold gilts strategies. Returns are calculated as continuously compounded percentage returns on a stock (the FTSE in the UK, the S&P500 in the US, the DAX in Germany) or on a long-term government bond. The profitability of the trading rules generated by the forecasts of the Markov switching model are found to be superior in gross terms compared with a simple buy-and-hold equities strategy. In the UK context, the former yields higher average returns and lower standard deviations. The switching portfolio generates an average return of 0.69% per month, compared with 0.43% for the pure bond and 0.62% for the pure equity portfolios. The improvements are not so clear-cut for the US and Germany. The Sharpe ratio for the UK Markov switching portfolio is almost twice that of the buy-and-hold equities portfolio, suggesting that, after allowing for risk, the switching model provides a superior trading rule. The improvement in the Sharpe ratio for the other two countries is, on the contrary, only very modest. To summarise The Markov switching approach can be used to model the gilt–equity yield ratio The resulting model can be used to produce forecasts of the probability that the GEYR will be in a particular regime Before transactions costs, a trading rule derived from the model produces a better performance than a buy-and-hold equities strategy, in spite of inferior predictive accuracy as measured statistically Net of transactions costs, rules based on the Markov switching model are not able to beat a passive investment in the index for any of the three countries studied.

10.8 Threshold Autoregressive Models Threshold autoregressive (TAR) models are one class of non-linear autoregressive models. Such models are a relatively simple relaxation of standard linear autoregressive models that allow for a locally linear approximation over a number of states. According to Tong (1990, p. 99), the threshold principle ‘allows the analysis of a complex stochastic system by decomposing it into a set of smaller sub-systems’. 597

The key difference between TAR and Markov switching models is that, under the former, the state variable is assumed known and observable, while it is latent under the latter. A very simple example of a threshold autoregressive model is given by (10.24). The model contains a first order autoregressive process in each of two regimes, and there is only one threshold. Of course, the number of thresholds will always be the number of regimes minus one. Thus, the dependent variable yt is purported to follow an autoregressive process with intercept coefficient μ1 and autoregressive coefficient ϕ1 if the value of the state-determining variable lagged k periods, denoted st−k is lower than some threshold value r. If the value of the state-determining variable lagged k periods, is equal to or greater than that threshold value r, yt is specified to follow a different autoregressive process, with intercept coefficient μ2 and autoregressive coefficient ϕ2. The model would be written (10.24) But what is st−k, the state-determining variable? It can be any variable that is thought to make yt shift from